2.2 Let the variable X have the binomial distribution, that is, p(x) = (") p*(q)"-*, x = 0,1,2,-,n If the mean of X is given to be u = np. Find E[X(X – 1)]. ii. Use results in (i) to find the variance of X. i.
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![2.2 Let the variable X have the binomial distribution, that is,
p(x) = (") p*(q)"-*, x= 0,1,2, -- ,n
If the mean of X is given to be µ = np.
Find E[X(X – 1)].
Use results in (i) to find the variance of X.
i.
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- 1. Let X be a random variable having pdf f(x) = 6x(1 – x) for 0 < I < 1 and 0 elsewhere. Compute the mean and variance of X. 2. Let X1, X2,..., X, be independent random variables having the same distribution as the variable from problem 1, and let X, = (X1+ ·.+Xn). Part a: Compute the mean and variance of X, (your answer will depend on n). Part b: If I didn't assume the variables were independent, would the calculation in part a still work? Or would at least part of it still work? 3. Suppose that X and Y are both independent variables, and that each has mean 2 and variance 3. Compute the mean and variance of XY (for the variance, you may want to start by computing E(X²Y²)). 4. Suppose that (X,Y) is a point which is equally likely to be any of {(0, 1), (3,0), (6, 1), (3, 2)} (meaning, for example, that P(X = 0 and Y = 1) = }). Part a: Show that E(XY) = E(X)E(Y). Part b: Are X and Y independent? Explain. 5. Let X be a random variable having a pdf given by S(2) = 2e-2" for 0Suppose that X1,..., X, is a random sample from a normal distribution with mean u and variance o?. Two unbiased estimators of o? are ởf = s° = E(x, - Xy°, and ô = (X1 - 2 Find the efficiency of ở relative to ôž.Consider an estimator of μ: W= 1/6 Y1+1/16Y2+1/4Y3+1/8Y4+1/2Y5 This is an example of a weighted average of the Yi's. Show that W is also an unbiased estimator of μ. Find the variance of W.5. Assume = 0.06 and let the force of mortality be given by 0.04, 0 < t < 20 { 0.05, 20let x has the p.d.f {2.xe-v2x 01.9.18. Find the mean and the variance of the distribution that has the cdf x <0 04. The standard deviation of X, denoted SD(X), is given by SD(X) = Var(X). Find SD(aX + b) if X has variance of o?.6. Show that 1 s2 E1(Xi – x)² is unbiased estimator of the population variance o? i=1 п-14. Let X - N(u, o²). The characteristic function (ch.f.) of X is given by $(w) = ejHw-o?w?/2, W E R, where j = v-1 is the imaginary unit. Use the ch.f. to find the mean and variance of X.Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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