Question

9) A fair coin is tossed n times, coming up Heads Nh times and Tails Nr = n – Nh times. Let Sn = Nh – Nt. Use Cramer's Theorem to show that for 0 < a < 1, 1-1/2 lim n-> P(Sn. = ( + (1 - a)1-a

Answer 1

Answer:

Given that:

A fair coin is tossed n times, coming up Heads NH times and Tails    times. Let

Use Cramer’s Theorem to show that for 0 < a < 1,

limn→∞ $\lim_{n\rightarrow \oe }P(S_n > an) ^{1/n} = [(1 + a) ^{1+a} (1 - a) ^{1-a }]^{-1/2}$

Let the number of heads be x times, then the number of tails will be n-x. The variable Sn then becomes x-(n-x) or 2x-n.

Now the probability of getting x heads in n tosses is ⁿC_x*(1/2)^x*(1/2)^(n-x) or ⁿC_x*(1/2)^n. and P(x>a) for a is a integer will be summation ⁿC_x*(1/2)^n from a+1 to n.

Thus for the expression P(S>a) can be written as

2x-n>a or x>(a+n)/2, thus the probability becomes summation ⁿC_x*(1/2)^n from integer((a+n)/2) to n.as a is a fraction less than 1, thus the int(a+n)/2) becomes simply int(n/2) or half the series, thus the summation is simply 2^(n-1).

Thus the expression becomes 1/2n

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