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Chapter 7: Q.7.36 (page 355)
Let be the number of and the number of that occur in rolls of a fair die. Compute .
The value ofis
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There are n items in a box labeled H and m in a box labeled T. A coin that comes up heads with probability p and tails with probability 1 鈭� p is flipped. Each time it comes up heads, an item is removed from the H box, and each time it comes up tails, an item is removed from the T box. (If a box is empty and its outcome occurs, then no items are removed.) Find the expected number of coin flips needed for both boxes to become empty. Hint: Condition on the number of heads in the first n + m flips.
Let Z be a standard normal random variable,and, for a 铿亁ed x, set
If are independent and identically distributed random variables having uniform distributions over , find
(a) ;
(b) .
If and Y are independent and identically distributed with mean and variance , find
Let X be a normal random variable with parameters 渭 = 0 and 蟽2 = 1, and let I, independent of X, be such that P{I = 1} = 1 2 = P{I = 0}. Now define Y by Y = X if I = 1 鈭扻 if I = 0 In words, Y is equally likely to equal either X or
(a) Are X and Y independent?
(b) Are I and Y independent?
(c) Show that Y is normal with mean and variance .
(d) Show that
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