Question

Draw up a table showing the mean, variance, and standard
deviation for the following three random variables, in terms of
*n*, *p*_{1} and *p*_{2},
where *n* is an even natural number.

*W is the number of successes in n trials with a probability
of success*

*X is the sum of the number of successes in n/2 trials with
probability of success p1 and the number of successes in n/2 trials
with probability of success p _{2}.*

Answer #1

1. we know that every alternative number is an even number so in n trails half of the numbers are even and its probability 0.5

If W is the no.of successes in n trails with probability p= 0.5

and its mean =0.5 n Variance = 0.25n Standard deviation = (0.25
n)^{1/2}

2. if X is the sum of the number of successes in n/2 trails with probability of success p1 and the no. successes in n/2 trails with probability p2 then X is a random variable with number of successes n/2 +n/2 = n with probability p1=p2=p = 0.5

its mean = 0.5 n variance =0.25 n standard deviation = (0.25
n)^{1/2}

3.if Y is the sum of the number of successes in n independent trails with probability p1=p2=0.5 then

its mean = (n+1) 0.5 variance =(n(n^{2} + 1)
-(n+1)^{2} ) 0.25 its S.D = ( variance )^{1/2
}

you can compare the standard deviation in each of the case by subtituting n=100 p1= 0.5 and p2 =0.5

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