Theorem 5-6. (b) Distribution of the Difference of Two Random Variables. If X and Y are independent continuous random variables, then the p.d.f. of U = X- Y, is given by : 00 = f(x, x + u) dx =| fx(x) fy (x + u) dx h(u) (5-27) %3D ...
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- The distribution of the random variable X is determined by the function: f(x) = ((x-1)2)/c ; for 1<x<3 = 0 ; inak Sketch the graph of f(x) Determine the FY(y) and fY(y) for Y = X^2 - 1The random variable X has a Bernoulli distribution with parameter p. A random sampleX1, X2, . . . , Xn of size n is taken of X. Show that the sample proportionX1 + X2 + · · · + Xnnis a minimum variance unbiased estimator of p.Given that X1, X2, . . . , Xn forms a random sample of size n from a geometric population withparameter p, show thatY =n∑j=1Let X1, ..., Xn be independent random variables with the distribution from the first problem "Suppose the lifetime of an appliance is a continuous random variable X. (i) If the support of fX is [0, 2] and its pdf f is linear with f(0) = 0, find the pdf. (ii) Find E[X] and explain in words what it represents. (iii) Find V ar(X) and explain in words what it represents". Let Yn = max{X1, ..., Xn}. ( a) What event in terms of X1, ..., Xn is equivalent to the event {Yn ≤ x}? (b) Find the cumulative distribution function (CDF) of Yn. (c) Find the probability density function (pdf) of Yn. (d) Find the expected value of Yn.
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