Let XBinnp X of successes in n trials Xn proportion of su

Let X~Bin(n,p). (X = # of successes in n trials, X/n = proportion of successes in n trials)

Let Pn = proportion of successes in n trials. Pn = X/n. Determine E(Pn), Var(Pn), Show Pn converges to P.

Solution

X~B(n,p)

E(Pn) = E(X/n)

= 1/n E(X)

= 1/n (np) = p

Thus E(Pn) = p

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2) Var(Pn)

= Var(x/n) = Var(x)/n²

= npq/n²

= pq/n

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Mean of Pn is p and in the long run as n becomes very large np = Mean becomes the mean of normal distribution.

Hence Mean/n will be a constant p

Hence it follows that Pn converges to P