Let X be a discrete random variable with the following...

Let X be a discrete random variable with the following PMF: p(x) = { 0.2 x = -1, 0.45 x = 0, 0.35 x = 1. Let X?n denote the sample mean of a random sample X1, . . . , Xn. (i) Suppose that n = 2. What is the sampling distribution of X?2? Provide its PMF. (ii) Suppose that n = 2. What is the sampling distribution of T = X1 % X2? Provide its PMF. (iii) Compute the standard error of X?n when n = 2. Explain what happens to the standard error when n changes from 2 to 3.

Transcript

00:01 Hello students, here from the question they given that the p of x which is 0 .2 for x is equal to minus 1, 0 .45 then for x is equal to 0 then 0 .35 for x is equal to plus 1.

00:18 Here x bar n denotes the sample mean of a random sample that is x1, x2 up to xn.

00:30 Therefore xn bar which is 1 by n into sum of i is equal to 1 to n that xi.

00:37 Now here the probability of x2 bar which is equal to minus 1 that is the probability of x1 is minus 1 and x2 which is minus 1 that is equal to 0 .04 and the probability of x2 bar which is minus 1 by 2 that is probability of x1 is equal to 0, x1 is equal to minus 1 that probability of x1 is equal to minus 1 and x2 which is equal to 0.

01:12 Therefore here by simplifying this we get our answer is equal to 0 .18 and also probability of x2 which is equal to 0 that is probability of x1 is equal to 1 and x2 which is minus 1 etc which is equal to 0 .3425 and probability of x2 which is equal to plus 1 by 2 that is equal to 0 .1575 plus 0 .1575 is equal to 0 .315 and also the probability of x2 bar which is equal to 1 is 0 .35 into 0 .25 we get 0 .1225 by now here the probability of x2 bar which is equal to x2 bar is equal to 0 .09 plus 0 .18 plus 0 .3425 plus 0 .315 plus 0 .1225 is equal to 1.

02:46 Therefore here the probability of x2 bar is equal to x2 bar that is equal to 0 .04 where x2 bar is equal to minus 1, 0 .18 the x2 bar which is equal to minus 1 by 2, 0 .3425 where x2 bar which is equal to 0, 0 .315 where x2 bar which is equal to 1 by 2 and then 0 .1225 that is x2 bar is equal to 1.

03:16 Now by summing all these values we get 1 so the answer is right and for the subpart 2, t is equal to x1 into x2 for n is equal to 2...

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