Question

Draw up a table showing the mean, variance, and standard deviation for the following three random variables, in terms of n, p₁ and p₂, 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 p₂.
Y is the sum of the number of successes in n independent trials, for each of which the probability of success is either p₁ with probability 0.5 or p₂ with probability 0.5.
Compare the standard deviations in the case that n=100, p1=0.1, and p2=0.5.

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(n² + 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