361, Math Problem set 5
Due 10/04/10 1. (1.6.8) Let X have the pmf p(x) = ( 1 )x , x = 1, 2, 3, . . . , and zero else2 where. Find the pmf of Y = X 3 . 2. (a) Pick a card from a standard deck. Let X denote the rank of the card(counting ace as one, J=11, Q=12, K=13.) Find the pmf of X . (b) Pick two cards from a deck, with replacement. Let Y denote the highest rank picked. Find the pmf of Y . 3. (1.7.8) A mode of a distribution of one random variable X is a value of x that maximizes the pdf or pmf. For X of the continuous type, f (x) must be continuous. If there is only one such x, it is called the mode of the distribution. Find the mode of each of the following distributions: p(x) (a) = ( 1 )x , x, 1, 2, 3, . . . , zero elsewhere. 2 (b) f (x) = 12x2 (1 x), 0 < x < 1, zero elsewhere. (c) f (x) = 1 x2 ex , 0 < x < , zero elsewhere. 2 4. (1.7.14) Let X have the pdf f (x) = 2x, 0 < x < 1, zero elsewhere. 3 Compute the probability that X is at least 4 given that X is at least 1 . 2 5. (1.7.17) Divide a line segment into two parts by selecting a point at random. Find teh probability that the larger segment is at least 3 times the shorter. Assume the point is chosen uniformly.
1 6. (1.7.22) Let X have the uniform pdf fX (x) = for < x < . Find 2 2 the pdf of Y = tan(X ). This is the pdf of a Cauchy distribution.
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