STA256H5S LEC0101-LEC0102, Assignment 2 Summer 2022
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STA256H5S LEC0101-LEC0102, Summer 2022
Assignment 2
Let X be a random variable. Show that
(a) If X is a discrete random variable that assigns positive probabilities to only the
positive integers, then
o
E(X) = P (X > k)
k_1
(b) If X is a continuous random variable whose support is x > 0 and with the cdf
F (x), then
o
E(X) = [1 _ F (x)]dx
(
Let X have the pmf p(x) = for k = 1, 2, 3, ..., k and zero elsewhere.
(a) Show that the mgf of X is
M (t) = ,
(b) Find E(X) both by the direct definition and mgf.
Let X be a continuous random variable with the pdf f (x) = e- lel, for x e R. Using Chebyshev’s inequality determine the upper bound for P (|X| > 5) and then compare it with the exact probability.
Let f (x, y) = e-e-亿 , 0 < x < o, 0 < y < o, and zero elsewhere, be the joint pdf of (X, Y). Then if Z = X + Y ,
(a) Find P (Z < 1) and P (Z < 3).
(b) Find P (Z < z) for 0 < z < o.
(c) Find the pdf of Z .
Let X1 and X2 have the joint pdf f (x1 , x2 ) = 2e-e1 -e2 for 0 < x1 < x2 < o, and zero elsewhere. Find the joint pdf of Y1 = 2X1 and Y2 = X2 _ X1 .
Let the conditional pdf of X1 given X2 = x2 be f (x1 |x2 ) = c1 x1 /x2(2) for 0 < x1 < x2 , 0 < x2 < 1, and zero elsewhere. Further, let the marginal pdf of X2 be f2 (x2 ) = c2 x2(4) for 0 < x2 < 1 ,and zero elsewhere. Then
(a) Determine the constants c1 and c2 .
(b) Find P ( < X1 < |X2 = ) and P ( < X1 < ).
Let X and Y have the joint pdf f (x, y) = 3x for 0 < y < x < 1 and zero elsewhere. Are X and Y independent? If not, find E(X|y).
2022-07-29