Reference no: EM132276104
Problem 1 - You have the conditional probability distribution, ΦY|x, of a continuous random variable, Y, given a known value of another random variable, X = x, whose marginal distribution, ΦX(x), is continuous and uniform in the range 0 ≤ x ≤ 10.
ΦY|x = xe-xy y > 0
Answer the following questions. If integration is required, just set up the integrals by clearly showing the integration variables, integrands and bounds. You do not evaluate the integrals to get full credit.
a) What is the conditional mean function, E[Y|x]?
b) What is the conditional variance function, var[Y|x]?
c) What is Pr(Y < 4|X = 4)?
d) What is the joint distribution of X and Y, ΦXY(x, y)?
e) What are the mean values of X and Y, E[X] and E[Y]?
f) What is the marginal distribution of Y, ΦY(y)?
g) What is the conditional distribution of X given Y = y?
h) What is the conditional mean function, E[X|y]?
i) What are the covariance and correlation of X and Y?
j) Are the random variables X and Y independent? Explain why or why not.
Problem 2 - The lifetimes of 6 components in an electronics device arc known to be independent random variables governed by exponential distributions with means of 1000, 2000, 3000, 4000, 7000 and 10,000 hours, respectively.
a) What is the probability that the lifetimes of all the components exceed 5000 hours?
b) What are the mean and variance of life of all of the components?
c) What is the probability that at least one component lifetime exceeds 10,000 hours?
d) If the electronics device continues to operate only if all components operate, what is the mean life of the electronics device?
Problem 3 - You are given the following probability density function, ΦX(x), for the cosine of the surface angle, X, of a laser etching tool. The distribution function has one parameter, α, and one constant, c.
ΦX(x) = (αx2-1)/c -1 ≤ x ≤ 1
a) What is the value of the constant, c?
b) What is the moment estimator for α?
c) Explain how you can determine if this moment estimator is unbiased.
d) Let S = (X1, . . . , X24) denote a random sample of sample size n = 24 with sample mean of -0.01 and sample variance of 0.1, for values of x. Use the moment estimator to estimate α?
e) Use the method of maximum likelihood to estimate the parameter α from a random sample. Just derive the equation, you do trot have to find a closed form expression or value for α.
f) Which method would you use to estimate α? Explain your answer.
Problem 4 - The results of 20 tests for levels of hemoglobin are reported in the following table.
|
15.3
|
14.9
|
16
|
14.8
|
16
|
15.7
|
15
|
14.6
|
14.4
|
15.3
|
|
15.7
|
15.6
|
16.2
|
14.6
|
16.2
|
14.5
|
16.2
|
15.7
|
14.4
|
15.2
|
Using this data, the sample mean is 15.33 and the sample variance is 0.382211.
a) Assume that the distribution underlying the sample data is normal, construct a 99% confidence interval on the mean. Show all work.
b) Construct a 99% prediction interval of a single future measurement of hemoglobin level.
c) Construct a 99% confidence interval on the standard deviation of hemoglobin level.
d) Are the results from parts a-c) reliable if the underlying distribution of the data is not normal? Explain why or why not.
Attachment:- Assignment File.rar