## Transcribed Text

Introduction to Statistical Inference
1. Wackerly, 4.130
Prove that the variance of a beta-distributed random variable with parameters a and 8
is
2
a8
2. Wackerly, 5.100
Let Z be a standard normal random variable and let Y1 = Z and Y2 =Z².
(a) What are E(Y1) and E(Y2)?
(b) What is E(Y1Y2)? [Hint: E(Y1Y2) = E(Z*).]
(c) What is Cou(Ys,Y2)?
(d) Show that Y/ and Y2 are dependent.
3. Wackerly, 6.7
Suppose that Z has a standard normal distribution.
(a) Find the density function of U = Z?.
(b) Does U have a gamma distribution? What are the values of a and ,8?
(c) What is another name for the distribution of U?
4. Wackerly, 6.15
Let Y have a distribution function given by
Find a transformation G(U) such that, if U has a uniform distribution on the interval
(0,1) G(U) has the same distribution as Y.
5. Let Y/ and Y2 be independent and uniformly distributed random variables on the interval
[0,0]. Find the
(a) probability density function of U = min(Y1,Y2) and U2 = max(Y1,Y2).
(b) mean and variance of U1.
(c) E(U2 - U1). Does it equal to E(U/)?

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