Suppose the random variable X has the following...

Suppose the random variable X has the following distribution: egin{tabular}{|c|c|c|c|c|} hline x & 2 & 4 & 6 & 10 \ hline P(X=x) & .3 & .35 & .25 & .1 \ hline end{tabular} (a) Calculate E(X) and V(X). (b) Suppose we are sampling a random sample of size two. Calculate the sampling distribution, the expected value, and variance, of the following statistics: i. sample mean ii. sample max iii. sample min iv. sample range. (c) Suppose we are sampling a random sample of size three. Calculate the sampling distribution, the expected value, and variance, of the following statistics: i. sample mean ii. sample max iii. sample min iv. sample range.

Transcript

00:02 Hello students, according to the given question we have to find the mean and variance values.

00:07 So, given x and probability of x values.

00:15 So, in the a bit expectation of x is equal to summation x into p of x.

00:22 So, that will be equal to 2 into 0 .3 plus 4 into 0 .35 plus 6 into 0 .25 plus 10 into 0 .1.

00:36 So, expectation of x is equal to 4 .5.

00:40 Now, we have to find expectation of x square that is equal to summation x square into p of x that will be equal to 2 square into 0 .3 plus 4 square into 0 .35 plus 6 square into 0 .25 plus 10 square into 0 .1.

01:05 So, that will be equal to 25 .3.

01:10 So, the variance of x is equal to expectation of x square minus expectation of x whole square that is equal to 25 .3 minus 4 .5 whole square that is equal to 5 .05.

01:26 In the b bit here the number of samples will be equal to 4 out of 2 that is 4 c 2 that is equal to 6.

01:39 So, here the samples expected value of x values and expectation of x square values variance of x values x minimum value and x maximum value x bar and range values.

01:57 So, the samples are 2 comma 4, 2 comma 6, 2 comma 10, 4 comma 6, 4 comma 10 and 6 comma 10.

02:15 So, the value of expectation of x is 2, 2 .1, 1 .6, 2 .9, 2 .4 and 2 .5 and expectation of x square is 6 .7, 10 .2, 11 .2, 14 .1, 15 .3, 15 .1, 19.

02:37 So, the variance is 2 .7, 5 .79, 8 .64, 5 .69, 9 .34, 12 .75.

02:51 The minimum is here 2.

02:54 So, the 2 of 4 the minimum is 2 and maximum is 4.

02:59 Here it is 2 and 6, here it is 2 and 10, 4, 10, 4 and 6, 4, 10 and 6 and 10...

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