Complete question is;
Loretta is rolling an unfair 6 sided die with a single number between 1 and 6 on each face. She has a 70% chance of rolling a four. She is playing a game with a friend and knows that if she rolls a four on three of her next five rolls she will lose the game. She wants to determine the probability that she rolls a four on three of her next five rolls.
Which simulation design has an appropriate device and a correct trial?
A) Using a fair coin let heads represent rolling a four and tails represent not rolling a four. Flip the coin five times.
B) Using a table of random digits select a digit between 0 and 9. Let 0-6 represent rolling a four and 7-9 represent not rolling a four. Select five digits.
C) Roll a fair die with a single digit between 1 and 6 on each face. Let four represent rolling a four and 1-3 and 5 and 6 represent not rolling a four. Roll the die five times.
D) Using a table of random digits select a digit between 1 and 6, ignoring digits 0, 7, 8, and 9. Let 4 represent rolling a four and 1-3 and 5 and 6 represent not rolling a four Select five digits.
Answer:
B) Using a table of random digits, select a digit between 0 and 9. Let 0-6 represent rolling a four and 7-9 represent not rolling a four. Select five digits.
Step-by-step explanation:
Since she knows that if she rolls a four on three of her next five rolls she will lose the game, then the best simulation that she will roll a four on three of the next five rolls will be option B because it uses a table of random digits and doesn't ignore any number but is well ordered with 0-6 representing a four and 7-9 not rolling a four.
A backpack is on sale for 30% off. If the sales price is $15.75, what is the original price? Show your work.
Answer:
$22.50
Step-by-step explanation:
As it is 30% off, $15.75 is 70% of the full price. You can then divide 15.75 by 70 [= 0.225] to get 1% of the full price. Multiply 0.225 by 100 to get the full price. 0.225 x 100 = 22.5
We can check that this answer is correct by multiplying 22.50 by 0.7. If you get 15.75 you know the answer is correct as 15.75 is 70% of the full price.
A sample of 76 body temperatures has a mean of 98.7. Assume that o is known to be 0.5 °F. Use a 0.05 significance level to test the claim that the mean body temperature of the population is equal to 98.5 °F, as is commonly believed. What is the value of test statistic for this testing?
The value of the test statistic in the scenario given above is 3.487
The test statistic can be calculated as follows:
t = (mean - hypothesized mean) / (standard deviation / √(sample size))
mean = 98.7
hypothesized mean = 98.5
standard deviation = 0.5
sample size = 76
Plugging these values into the formula, we get the following test statistic:
t = (98.7 - 98.5) / (0.5 / √(76)) = 3.487
Therefore, the value of the test statistic in the above scenario is 3.487
Learn more on test statistic: https://brainly.com/question/15110538
#SPJ4
Lizzy had test scores of: 72, 94, 108, 60What is the RANGE of her test scores?
Answer:
48
Step-by-step explanation:
To find the range, you have to subtract the biggest/largest number by the smallest number:
108 - 60 = 48
The range is 48.
The number of peopie that have "liked" Mindy's Candy Store web
site can be represented by y = 270(1.65), where t is the number of weeks after a
review in a national magazine. Interpret the parameters of the equation. Make a
prediction about the future number of likes on the web site.
Answer:
Kindly check explanation
Step-by-step explanation:
Given that, the number of people who liked Mindy's website is given by :
270(1.65)^t ; t = number of weeks after a review
Interpreting the equation :
The equation represents an exponential growth function :
y = a(1 + r)^t
a = initial amount ; r = growth rate ; t = period
Comparing the function with the equation given ;
270 = a = number of people who liked the website before the review
1 + r = 1.65
r = 1.65 - 1
r = 0.65 ; 0.65 * 100% = 65%
Site grew at a rate of 65% per week after review
Ther predicted number of likes 2 weeks after the review :
y = 270(1.65)^2
y = 270 * 2.7225
y = 735.075
Hence, number of likes after 2 weeks = 735
If cot θ= tan (pi/6) what is cos θ? Round your answer to the nearest hundredth.
Answer:3.46
Step-by-step explanation:
A brewery produces a particular type of beer. If the process is running well then the probability that a randomly selected bottle is of good quality is 90%, independently of the quality of any of the other bottles. If the process is not running well then that probability is only 60%. Without further information, the probability that the process is running well is 80%. (a) 10 bottles are produced. Find the expected number of good quality bottles in this batch. (b) of this batch, the first 4 bottles are tested, and all are of good quality except for the second bottle. What is the probability that the process is running well? (c) The 5th bottle is now tested as well. What is the probability that this bottle is of good quality?
(a) The expected number of good quality bottles in this batch is 7.2
(b) The probability that the process is running well, given that the second bottle is not of good quality, is 0.5
(c) The probability that the 5th bottle is of good quality is 0.84
(a) To find the expected number of good quality bottles in the batch of 10 bottles, we can multiply the probability of each bottle being of good quality by the total number of bottles.
Given that the probability of a bottle being of good quality is 90% if the process is running well and 60% if the process is not running well, and the probability that the process is running well is 80%, we can calculate the expected number of good quality bottles as follows:
Expected number of good quality bottles = (Probability of running well) × (Probability of good quality | running well) × (Total number of bottles)
= (0.8) × (0.9) × (10)
= 7.2
Therefore, the expected number of good quality bottles in this batch is 7.2.
(b) Given that the first 4 bottles tested are all of good quality except for the second bottle, we need to find the probability that the process is running well.
Let A be the event that the process is running well, and B be the event that the second bottle is not of good quality.
Using Bayes' theorem, we can calculate the probability:
P(A | B) = (P(B | A) × P(A)) / P(B)
P(B | A) is the probability that the second bottle is not of good quality given that the process is running well, which is 1 - 0.9 = 0.1.
P(A) is the probability that the process is running well, which is given as 0.8.
P(B) is the probability that the second bottle is not of good quality. To calculate this, we need to consider the cases where the process is running well and not running well.
P(B) = P(B | A) × P(A) + P(B | not A) × P(not A)
= 0.1 × 0.8 + 0.4 × 0.2
= 0.08 + 0.08
= 0.16
Now, we can calculate the probability using Bayes' theorem:
P(A | B) = (0.1 × 0.8) / 0.16
= 0.5
Therefore, the probability that the process is running well, given that the second bottle is not of good quality, is 0.5.
(c) Now, if the 5th bottle is tested as well, we want to find the probability that this bottle is of good quality.
Let C be the event that the 5th bottle is of good quality.
Using the law of total probability, we can calculate the probability:
P(C) = P(C | A) × P(A) + P(C | not A) × P(not A)
= 0.9 × 0.8 + 0.6 × 0.2
= 0.72 + 0.12
= 0.84
Therefore, the probability that the 5th bottle is of good quality is 0.84.
To learn more about probability
https://brainly.com/question/13604758
#SPJ11
HELP NOW!!! 100 POINTS! Cylinder A has a radius of 10 inches and a height of 5 inches. Cylinder B has a volume of 750π. What is the percentage change in volume from cylinder A to cylinder B?
50% decrease
75% decrease
50% increase
200% increase
Answer:
50% increase
Step-by-step explanation:
The percentage change in volume from cylinder A to cylinder B is 50% volume increased by 50% option (C) is correct.
What is a cylinder?In geometry, it is defined as the three-dimensional shape having two circular shapes at a distance called the height of the cylinder.
We know the formula for the volume of the cylinder is given by:
[tex]\rm V = \pi r^2h[/tex]
We have r = 10 inches and h = 5 inches
[tex]\rm V = \pi \times10^2\times5[/tex]
V = 500π cubic inches
Percent change in volume from cylinder A to cylinder B:
[tex]=\rm \frac{750\pi-500\pi}{500\pi} \times100[/tex]
= 50%
Thus, the percentage change in volume from cylinder A to cylinder B is 50% volume increased by 50% option (C) is correct.
Learn more about the cylinder here:
brainly.com/question/3216899
#SPJ2
HELP ASAP PLEASE ILL MARK BRAINLIEST
How far from the tower can it be placed, to the nearest foot?
Answer:
43 feet would be the correct answer.
Step-by-step explanation:
In circle O, radius OQ measures 9 inches and arc PQ measures 6π inches. Circle O is shown. Line segments P O and Q O are radii with length of 9 inches. Angle P O Q is theta. What is the measure, in radians, of central angle POQ? StartFraction 2 pi Over 3 EndFraction radians StartFraction 3 pi Over 4 EndFraction centimeters StartFraction 4 pi Over 3 EndFraction radians StartFraction 3 pi Over 2 EndFraction radians
The answer is A. 2pi/3 radians.
Answer:
First Option
Explanation:
A) 2pi/3 radians
Hope this helps :)
A bag contains 9 and 54 blue marbles. If a representative sample contains 3 white marbles, then how many blue marbled would you expect to contain? Explain
Answer:
There should be 18 blue marbles in the representative sample.
Step-by-step explanation:
Given that a bag contains 9 white and 54 blue marbles, if a representative sample contains 3 white marbles, to determine how many blue marbles would you expect to contain the following calculation must be performed:
9 = 3
54 = X
54 x 3/9 = X
162/9 = X
18 = X
Therefore, there should be 18 blue marbles in the representative sample.
12 = a (-6 + 5) (-6 - 6)
Answer:
a=1
Step-by-step explanation:
In a right triangle, one acute angle measure 35 degrees and its adjacent side is 40 yards.
Find the opposite side.
Paragraph. Describe how to find the opposite side of one right triangle with one acute angle of 35 degrees and adjacent side of 40 yards.
Word Bank: right triangle, opposite and adjacent side,
Answer:
Opposite side =28 yards
Step-by-step explanation:
Tan_°=Opposite÷ Adjacent
Tan 35° =Opposite÷40yards
Opposite=40*Tan35°
Opposite=28yards
Consider the following equation: 4 + 6x = 6x + 4. Explain why the equation has many solutions.
I NEED HELP ASAPPPP ILL GIVE BRAINLIESTTTTTT
2/3 x 6 need help asap
Answer:
4
Step-by-step explanation:
Answer:
4
Step-by-step explanation:
2/3 x 6 is 4 because when you times the 2 in 2/3 u get 12 and 12/3 is the same as 4
3/4 + 7/8 + 5/6
(this is here because the question was too short)
Answer:23/8Step-by-step explanation:(LCM of 4,8 and 6 = 24 )
3×6/4×6 + 7×3/8×3 + 5×4/6×4 28+21+20/24
59/24
I hope you got it❤️❤️❤️And thanks your patience ❤️.SORRY
A recent study sought to determine if exercising after studying for an exam may help retain information. The 150 student volunteers that participated in the study were randomly assigned to three groups. All students were given a picture memory task that they studied for 15 minutes. After studying, one of the groups was instructed to exercise for 30 minutes immediately after studying. A second group was instructed to wait 1 hour after studying and then exercise for 30 minutes. The third group was instructed to not exercise after studying. The next day all 150 student volunteers were given a test to determine how well they could remember what they studied. Here are the results:
Answer:
The correct answer is (A).
Step-by-step explanation:
Because the data came from 3 groups in a randomized experiment, the appropriate test is a chi-square test for homogeneity.
whats is the answer to this equation?
who ever gets the right answer gets 25 points.
Answer:
51.2 m
Step-by-step explanation:
To find perimeter add all sides up
Suppose that 7% of the Karak tea packs produced by the company Chai Karak are defective. A shipment of 10,000 packs is sent to Ishbeliyah co-op. The co-op inspects a Simple Random Sample (SRS) of 10 packs. Let X = number of defective Karak tea packs in the SRS of size 10.
What is the probability that none of the packs are defective P(X = 0)?
What is the probability that 5 packs are defective?
What is the probability that all the packs are defective?
What is the probability that 7 or more packs are defective?
The probability that none of the packs are defective is 0.478. The probability that five packs are defective is 0.000455.The probability that all the packs are defective is 2.8243e-14.The probability that 7 or more packs are defective is 0.00416 (approx).
A shipment of 10,000 Karak tea packs is produced by the company Chai Karak. If 7% of the packs are defective, what is the probability that: none of the packs are defective, five packs are defective, all the packs are defective, and seven or more packs are defective? The number of trials, n, is 10 and the probability of a defective tea pack is 0.07. Therefore, the number of successful trials, X, follows a binomial distribution. Formula for binomial distribution: P(X = k) = nCk × pk × (1 − p)n−kWhere nCk = number of combinations of n things taken k at a time = n! / (k! (n-k)!)a. The probability that none of the packs are defective P(X = 0):P(X = 0) = nC0 * p0 * (1-p)n-0= 10C0 * 0.07^0 * (1-0.07)^10= 1 * 1 * 0.478= 0.478Therefore, the probability that none of the packs are defective is 0.478.
The probability that 5 packs are defective:P(X = 5) = nC5 * p^5 * (1-p)n-5= 10C5 * 0.07^5 * (1-0.07)^5= 252 * 0.0000028 * 0.649= 0.000455Therefore, the probability that five packs are defective is 0.000455.
The probability that all the packs are defective:P(X = 10) = nC10 * p^10 * (1-p)n-10= 10C10 * 0.07^10 * (1-0.07)^0= 1 * 2.8243e-14 * 1= 2.8243e-14Therefore, the probability that all the packs are defective is 2.8243e-14.
The probability that 7 or more packs are defective: P(X ≥ 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)P(X = 7) = nC7 * p^7 * (1-p)n-7= 10C7 * 0.07^7 * (1-0.07)^3= 120 * 0.0000953677 * 0.657= 0.00416P(X = 8) = nC8 * p^8 * (1-p)n-8= 10C8 * 0.07^8 * (1-0.07)^2= 45 * 0.0000024969 * 0.859= 0.000011P(X = 9) = nC9 * p^9 * (1-p)n-9= 10C9 * 0.07^9 * (1-0.07)^1= 10 * 0.00000005 * 0.93= 4.65e-7P(X = 10) = nC10 * p^10 * (1-p)n-10= 10C10 * 0.07^10 * (1-0.07)^0= 1 * 2.8243e-14 * 1= 2.8243e-14P(X ≥ 7) = 0.00416 + 0.000011 + 4.65e-7 + 2.8243e-14= 0.00416Therefore, the probability that 7 or more packs are defective is 0.00416 (approx).
Learn more about probability here:
https://brainly.com/question/31828911
#SPJ11
If (81)(27)2x-5-93-4⁰, then x = ________
The value of x that satisfies the equation (81)(27)^(2x-5) - 9^(3-4x) = 0 is x = 17/14.
To find the value of x in the equation (81)(27)^(2x-5) - 9^(3-4x) = 0, we can use the properties of exponents and logarithms to simplify and solve the equation. By equating the bases and exponents on both sides, we can determine the value of x.
We start by simplifying the equation. Applying the exponent properties, we have (3^4)(3^3)^(2x-5) - (3^2)^(3-4x) = 0.
Simplifying further, we get (3^(4 + 3(2x-5))) - (3^(2(3-4x))) = 0.
Using the property (a^b)^c = a^(b*c), we can rewrite the equation as 3^(4 + 6x - 15) - 3^(6 - 8x) = 0.
Combining like terms, we have 3^(6x - 11) - 3^(6 - 8x) = 0.
To equate the bases and exponents, we set 6x - 11 = 6 - 8x.
Simplifying the equation, we get 14x = 17.
Dividing both sides by 14, we find that x = 17/14.
Therefore, the value of x that satisfies the equation (81)(27)^(2x-5) - 9^(3-4x) = 0 is x = 17/14.
Learn more about exponents here:
https://brainly.com/question/5497425
#SPJ11
Can the following side measures make a triangle: 11in., 12in. and 13in?
Answer:
Yes
Step-by-step explanation:
Hello There!
Remember the sum of the two shorter sides has to be greater than the largest side in order for it to be a triangle
11+12 must be greater than 13
11+12=23
23>13 therefore the side lengths can form a triangle
Draw a graph which has the following adjacency matrix: [2 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 (b) Label the vertices of your graph A, B, C, D (in any order). (c) How many different paths of length 10 are there from the vertex you labelled A to the vertex you labeled D in the graph you draw in part (a)?
Given an adjacency matrix, we need to find the graph which has the following adjacency matrix:[2 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0]We know that an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.The diagonal elements of the matrix are always zero since self-loops are not allowed in a simple graph.The (i, j)-entry of the matrix is 1 if the ith vertex and jth vertex are adjacent, and 0 otherwise. Hence, for the given matrix, there is an edge between vertex 1 and vertex 2 and an edge between vertex 1 and vertex 3. Similarly, there are other edges between vertices.Here is the graph which has the given adjacency matrix:Graph with adjacency matrixFrom the graph above, we can label the vertices A, B, C, D in any order. Let's say we label the vertices as A = vertex 1, B = vertex 2, C = vertex 3, and D = vertex 4.Now, we need to find the number of different paths of length 10 from vertex A to vertex D.The number of paths of length n from vertex i to vertex j in a graph is given by the (i, j)-entry in the n-th power of the adjacency matrix. Here, we need to find the number of paths of length 10 from vertex 1 to vertex 4. So, we need to compute the 10th power of the given adjacency matrix:[2 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0] ^ 10We can use a calculator or software like MATLAB to compute the matrix power. After computing, we get that the (1, 4)-entry of the 10th power of the matrix is 8. Therefore, there are 8 different paths of length 10 from vertex A to vertex D.
Learn more about adjacency here: brainly.com/question/22880085
#SPJ11
An airplane's altitude changed -378 feet over 77 minutes. What was the mean change of altitude in feet per minute?
Answer:
-4.91 feet
Step-by-step explanation:
Mean is a measure of central tendency. It is the average of a set of numbers. It is determined by adding the numbers together and dividing it by the total number
Mean = Total change in altitude / total time
378 / 77 = 4.91
Help plssss I need to find mr pryor account balance
Answer:
$1,462.50
Step-by-step explanation:
We are to find the total amount after 4 years
Principal => $1,250
Rate => 4.25%
The formula to find the total amount =
A = P(1 + rt)
First, converting R percent to r a decimal
r = R/100 = 4.25%/100 = 0.0425 per year.
Solving our equation:
A = 1250(1 + (0.0425 × 4)) = 1462.5
A = $1,462.50
The total amount accrued, principal plus interest, from simple interest on a principal of $1,250.00 at a rate of 4.25% per year for 4 years is $1,462.50.
There are 12 boys and 16 girls in a classroom. Which represents the simplified ratio of girls to students in the classroom?
3 to 4
4 to 3
4 to 7
7 to 4
Answer:
3 to 4
Step-by-step explanation:
12÷4=3
16÷4=4
so that makes it 3 to 4
Suppose the prevalence of is 12.5%. We assume the
diagnostic test has a sensitivity of 80% and a
95% specificity. What is the probability of getting a negative
result?
The probability of getting a negative result is 0.175 or 17.5%.
To calculate the probability of getting a negative result, we need to consider the sensitivity and specificity of the diagnostic test.
Given that the prevalence of the condition is 12.5%, we can assume that 12.5% of the population has the condition, and the remaining 87.5% does not.
The sensitivity of the test is 80%, which means that it correctly identifies 80% of the individuals with the condition as positive.
The specificity of the test is 95%, which means that it correctly identifies 95% of the individuals without the condition as negative.
To calculate the probability of getting a negative result, we need to consider both the true negative rate (1 - sensitivity) and the proportion of individuals without the condition (1 - prevalence).
Probability of getting a negative result = (1 - sensitivity) × (1 - prevalence)
= (1 - 0.80) × (1 - 0.125)
= 0.20 * 0.875
= 0.175
Therefore, the probability of getting a negative result is 0.175 or 17.5%.
Learn more about probability https://brainly.com/question/31828911
#SPJ11
Can someone explain why the answer is True. Will make brainliest.
Answer:
True
Step-by-step explanation:
With Sin and Cos, the rule is that the opposites are the same:
SinA=CosB
CosB=SinA
In this problem, it is showing SinA=cosB, so it is the same, it is true.
Hope this helps!
What Is The Improper fraction for 6/7 and 4/5
Answer:
11 upon 4 is your 6 upon 4 answer
Suppose that X, Y and Z are three jointly normally distributed random variables with E[X] = 0, E[Y] = 1, E[Z] = 2 and the variance-covariance martrix of (X, Y, Z) is 10 0 1 Var [] = [] 10 2 1 2 10 (i) Estimate X given that Y = 0.5 and Z = -3 using an unbiased minimum variance estimator. (ii) Determine the variance of the above estimator. (b) IntelliMoto is car manufacturer that produces vehicles equipped with a fault detection system that uses information from various sensors to inform the driver about possible faults in the braking system. The system diagnoses faults correctly with probability 99%, but gives false alarms with probability 2%. It is known that such faults occur with probability 0.05%. If the system diagnoses a fault, what is the probability a fault has actually occured?
(i) Estimate X given that Y = 0.5 and Z = -3 using an unbiased minimum variance estimator.
Estimate of X given that Y = 0.5 and Z = -3 can be obtained by applying the conditional expectation formula, E[X|Y=y, Z=z], where y=0.5 and z=-3.E[X|Y=y, Z=z] = E[X] + Cov[X,Y]/Var[Y] * (Y - E[Y]) + Co v[X,Z]/Var[Z] * (Z - E[Z])E[X|Y=0.5, Z=-3] = 0 + (0/10) * (0.5 - 1) + (1/2) * (-3 - 2) = -2, which is the unbiased minimum variance estimator.(ii) Determine the variance of the above estimator.
The variance of the unbiased minimum variance estimator is given by Var[X|Y=y, Z=z] = Var[X] - Cov[X,Y]^2/Var[Y] - Cov[X,Z]^2/Var[Z] + 2Cov[X,Y]Cov[X,Z]/(Var[Y]*Var[Z])Var[X|Y=0.5, Z=-3] = 10 - 0^2/10 - 1^2/2 + 2(0)(1)/(10*2) = 9.75 (b)
Intelli Moto is car manufacturer that produces vehicles equipped with a fault detection system that uses information from various sensors to inform the driver about possible faults in the braking system. The system diagnoses faults correctly with probability 99%, but gives false alarms with probability 2%. It is known that such faults occur with probability 0.05%. If the system diagnoses a fault, what is the probability a fault has actually occured?
The probability of a fault actually occurring is P(Fault) = 0.05%, which is the prior probability of a fault.
The probability of a correct diagnosis is P(Diagnosis | Fault) = 99%, which is the probability of a positive test result given that a fault has actually occurred.
The probability of a false alarm is P(Diagnosis | No Fault) = 2%, which is the probability of a positive test result given that no fault has actually occurred.
The probability of a positive test result isP(Diagnosis) = P(Fault)*P(Diagnosis | Fault) + P(No Fault)*P(Diagnosis | No Fault)= 0.05% * 99% + 99.95% * 2% = 2.039%.The probability of a fault given a positive test result can be obtained by Bayes' theorem,P(Fault | Diagnosis) = P(Diagnosis | Fault)*P(Fault)/P(Diagnosis)= 99% * 0.05% / 2.039% = 2.43%, which is the probability a fault has actually occurred given that the system diagnoses a fault.
The probability that a fault has actually occurred given that the system diagnoses a fault is 2.42%.
(i) To estimate X given that Y = 0.5 and Z = -3 using an unbiased minimum variance estimator, we need to determine the distribution of X | Y = 0.5, Z = -3 and use the formula for conditional expectation of a jointly normally distributed random variable. The distribution of X | Y = 0.5, Z = -3 is also normal since it is a conditional distribution of a jointly normally distributed random variable. To find the mean of the distribution, we use the formula for conditional expectation:
[tex]E[X | Y = 0.5, Z = -3] = E[X] + Cov[X, Y | Z = -3] (Y - E[Y | Z = -3]) / Var[Y | Z = -3] + Cov[X, Z | Y = 0.5] (Z - E[Z | Y = 0.5]) / Var[Z | Y = 0.5][/tex]
where Cov[X, Y | Z = -3] is the conditional covariance of X and Y given Z = -3,
E[Y | Z = -3] is the conditional mean of Y given Z = -3,
Var[Y | Z = -3] is the conditional variance of Y given Z = -3,
Cov[X, Z | Y = 0.5] is the conditional covariance of X and Z given Y = 0.5,
and E[Z | Y = 0.5] and Var[Z | Y = 0.5] are the conditional mean and variance of Z given Y = 0.5 respectively.
We are given that
E[X] = 0, E[Y] = 1, E[Z] = 2,
Var[X] = 10, Var[Y] = 2, Var[Z] = 1,
and Cov[X, Y] = Cov[X, Z] = Cov[Y, Z] = 0.
Also, Y = 0.5 and Z = -3.
Hence, we have:
[tex]Cov[X, Y | Z = -3] = Cov[X, Y] / Var[Z] = 0[/tex],
[tex]E[Y | Z = -3] = E[Y] =[/tex]1,
[tex]Var[Y | Z = -3] = Var[Y] = 2[/tex],
[tex]Cov[X, Z | Y = 0.5] = Cov[X, Z] / Var[Y] = 0[/tex].
The conditional mean of Z given Y = 0.5 is given by
[tex]E[Z | Y = 0.5] = E[Z] + Cov[Y, Z] (Y - E[Y]) / Var[Y] = 2 + 0.5 (0 - 1) / 2 = 1.5.[/tex]
The conditional variance of Z given Y = 0.5 is given by
[tex]Var[Z | Y = 0.5] = Var[Z] - Cov[Y, Z]^2 / Var[Y] = 1 - 0^2 / 2 = 1[/tex].
Hence, the mean of the distribution of X | Y = 0.5, Z = -3 is:
[tex]E[X | Y = 0.5, Z = -3] = 0 + 0 (0.5 - 1) / 2 + 0 (-3 - 1.5) / 1 = -0.75[/tex]
To find the variance of the unbiased minimum variance estimator, we use the formula for conditional variance of a jointly normally distributed random variable:
[tex]Var[X | Y = 0.5, Z = -3] = Var[X] - Cov[X, Y | Z = -3]^2 / Var[Y | Z = -3] - Cov[X, Z | Y = 0.5]^2 / Var[Z | Y = 0.5][/tex]
where Var[X], Cov[X, Y | Z = -3], and Cov[X, Z | Y = 0.5] are given above,
and Var[Y | Z = -3] and Var[Z | Y = 0.5] are calculated as follows:
[tex]Var[Y | Z = -3] = Var[Y] - Cov[X, Y]^2 / Var[Z] = 2 - 0^2 / 1 = 2Var[Z | Y = 0.5] = Var[Z] - Cov[Y, Z]^2 / Var[Y] = 1 - 0^2 / 2 = 1[/tex]
Hence, we have:
[tex]Var[X | Y = 0.5, Z = -3] = 10 - 0^2 / 2 - 0^2 / 1 = 10[/tex]
(ii) The variance of the unbiased minimum variance estimator is Var[X | Y = 0.5, Z = -3] = 10.
(b) Let A denote the event that a fault has actually occurred, D denote the event that the system diagnoses a fault,
P(A) = 0.05%, P(D | A) = 99%, and P(D | A') = 2%, where A' is the complement of A.
We need to find P(A | D), the probability that a fault has actually occurred given that the system diagnoses a fault.
By Bayes' theorem, we have:
[tex]P(A | D) = P(D | A) P(A) / P(D)[/tex]
where P(D) is the total probability of the system diagnosing a fault, which is:
[tex]P(D) = P(D | A) P(A) + P(D | A') P(A') = 0.99 (0.0005) + 0.02 (1 - 0.0005) = 0.0205[/tex]
Hence, we have:
[tex]P(A | D) = 0.99 (0.0005) / 0.0205 = 0.0242[/tex] or 2.42%
Therefore, the probability that a fault has actually occurred given that the system diagnoses a fault is 2.42%.
To know more about probability, visit:
https://brainly.com/question/31828911
#SPJ11