5 Let the random variable Yn have a binomial distribution that is bin(n, p). (a) Prove that Yn/n converges in probability to p. This result is one form of the weak law of large numbers. (b) Prove that 1 - Yn/n converges in probability to 1 - p. (c) Prove that(Yn/n)(1 - Yn/n) converges in probability to p(1 - p)
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To do this, we use the fact that p=binomial(n,k). Now, since Yn/n is a random variable, we can use the law of averages to calculate the following: (n-k+1)/2=binomial(n-k+1,k) =binomial(n-1,k+1)+binomial(n-2,k+1) =binomial(n-3,k+1)+binomial(n-4,k+1) Show more…
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