5. Let X, and X, be discrete random variables with joint pdf given by the following table: X2 1 12 0. 1 X1 18 4 15 a) Find the marginal pdf's of X, and X, b) Are X, and X, independent? Why or why not? c) Find P(X, < X2)
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- 5. let X1 and X2 have the joint pdf f(x1,x2) = x1+ x2, 0 < x1 < 1, 0 < x2 < 1. Find the conditional mean and variance of X2 given X1 = x1, 0 < x1 < 1.Suppose that X1 and X2 are independent and exponentially distributed random vari- 1 ables, both with mean . Let Y1 = X1 + X2 and Y2 X1 Find the joint pdf of X2 Y1 and Y2 (including its domain).10. Let and be independent random variables representing the lifetime (in 100 hours) of Type A and Type B light bulbs, respectively. Both variables have exponential distributions, and the mean of X is 2 and the mean of Y is 3. a) Find the joint pdf f(x, y) of X and Y. b) Find the conditional pdf f₂ (ylx) of Y.. c) Find the probability that a Type A bulb lasts at least 300 hours and a Type B bulb lasts at least 400 hours. d) Given that a Type B bulb fails at 300 hours, find the probability that a Type A bulb lasts longer than 300 hours. e) What is the expected total lifetime of two Type A bulbs and one Type B bulb?
- Let X1 and X2 have the joint pdf f(x1,x2) = x1+x2, 0 < x1 <1 and and variance 0 < x2 <1. Find the conditional mean of X2, given X1 = x1.5) Let Y be binomial (n = 4, II = .3) (a) Calculate the mean and variance by filling in the following table: f(yi) Xi X f(Yi) Yi2 X f(yi) b) Calculate the mean and variance of Y using formulas.B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.
- 13. Let Y,, Y2, ..., Yn denote a random sample from a population with pdf f(yl0) = (0 + 1)yº,0 -1 a) Find an estimator for theta by the method of moments. b) Find the maximum likelihood estimator for theta.Let X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.