1.12 In n rolls of a fair die, let X be the number of times 1 is rolled, and Y the number of times 2 is rolled. Find the conditional distribution of X given Y = y
1.12 In n rolls of a fair die, let X be the number of times 1 is rolled, and Y the number
of times 2 is rolled. Find the conditional distribution of X given Y = y.
1.13 Random variables X and Y have joint density
f(x, y) = 3y, for 0 < x < y < 1.
(a) Find the conditional density of Y given X = x.
(b) Find the conditional density of X given Y = y. Describe the conditional
distribution.
1.14 Random variables X and Y have joint density function
f(x, y) = 4e−2x
, for 0 < y < x < ∞.
(a) Find the conditional density of X given Y = y.
(b) Find the conditional density of Y given X = x. Describe the conditional
distribution.
1.15 Let X and Y be uniformly distributed on the disk of radius 1 centered at the
origin. Find the conditional distribution of Y given X = x.
1.16 A poker hand consists of five cards drawn from a standard 52-card deck. Find
the expected number of aces in a poker hand given that the first card drawn is
an ace.
1.17 Let X be a Poisson random variable with ? = 3. Find E(X|X > 2).
1.18 From the definition of conditional expectation given an
E(IB|A) = P(B|A).
1.19 See Example 1.21. Find the variance of the number of flips needed to get two
heads in a row.
1.20 A fair coin is flipped repeatedly.
(a) Find the expected number of flips needed to get three heads in a row.
(b) Find the expected number of flips needed to get k heads in a row.
1.21 Let T be a nonnegative, continuous random variable. Show
E(T) = ∫
∞
0
P(T > t) dt.
1.22 Find E(Y|X) when (X, Y) is uniformly distributed on the following regions.
(a) The rectangle [a, b]×[c, d].
(b) The triangle with vertices (0, 0),(1, 0), (1, 1).
(c) The disc of radius 1 centered at the origin.
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