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Chapter 6: Q. 6.27 (page 273)
If are independent exponential random variables with respective parameters and , find the distribution of . Also compute .
The distribution of Z is
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Let X1, ... , Xn be independent uniform (0, 1) random variables. Let R = X(n) 鈭� X(1) denote the range and M = [X(n) + X(1)]/2 the midrange of X1, ..., Xn. Compute the joint density function of R and M.
Suppose that n points are independently chosen at random on the circumference of a circle, and we want the probability that they all lie in some semicircle. That is, we want the probability that there is a line passing through the center of the circle such that all the points are on one side of that line, as shown in the following diagram:
Let P1, ... ,Pn denote the n points. Let A denote the event that all the points are contained in some semicircle, and let Ai be the event that all the points lie in the semicircle beginning at the point Pi and going clockwise for 180鈼�, i = 1, ... , n.
(a) Express A in terms of the Ai.
(b) Are the Ai mutually exclusive?
(c) Find P(A).
Let W be a gamma random variable with parameters (t, 尾), and suppose that conditional on W = w, X1, X2, ... , Xn are independent exponential random variables with rate w. Show that the conditional distribution of W given that X1 = x1, X2 = x2, ... , Xn = xn is gamma with parameters t + n, 尾 + n i=1 xi .
Suppose that X, Y, and Z are independent random variables that are each equally likely to be either 1 or 2. Find the probability mass function of
(a) ,
(b) , and
(c)
Suppose that balls are chosen without replacement from an urn consisting of white andred balls. Let role="math" localid="1649430608157" equal if the th ball selected is white, and let it equal otherwise. Give the joint probability mass function of
(a) ;
(b) .
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