1. Find the probability mass function (pmf) for a random variable with the following moment generating function (mgf),
M (t) = 1/2e-1 + 1/4 + 1/5 et +1/20e5t
2. Complete the following integrals to obtain simplified functions of t, and determine the range of t for which each of these integrals is defined:
a. MM (tt) = ∫1 ee xxtt dddd
b. MM(tt) = ∫∞ ee xx(tt - 1) dddd
c. Determine the probability density functions (pdfs) corresponding to the moment generating functions in parts a) and b).
3. Consider the following joint probability distribution pp(dd, yy) for random variables X and Y:
|
x \ y
|
0
|
1
|
2
|
|
|
0
|
0.10
|
0.15
|
0.05
|
0.30
|
|
1
|
0.10
|
0.20
|
0.40
|
0.70
|
|
|
0.20
|
0.35
|
0.45
|
|
a. Find PP(XX = YY)
b. Find PP(YY > 0 | XX > 0)
c. Find EE(XX + YY)
d. Find EE(YY|XX = 1)
4. Suppose X and Y are jointly distributed random variables with density,ffxxxx (dd, yy) = CC, 0 < dd < yy < 1
a. Sketch the support for X and Y.
b. Find CC.
c. Find PP ???? > ????.
5. Let ffxx,xx (dd, yy) = ee - xx - yy , 0 < dd < ∞, 0 < yy < ∞ be the pdf of X and Y. a. Let ZZ = XX + YY. Compute PP(ZZ ≤ 0).
a. Let ZZ = XX + YY. Compute PP(ZZ ≤ 0).
b. Find the pdf of Z.
c. Find MMXX,XX ttxx , ttyy.