Question

Suppose you toss a fair coin 10 times resulting in a sequence of heads (H) and tails (T). Let X be the number of times that the sequence HT appears. For example, HT appears thrice in

THTHHTHHHT Find E(X).

Use Indicator random variables.

Answer 1

Let us define 9 indicator random variables as:

for i = 1(1)9

P(HT) = P(H) P(T) = 0.5 * 0.5 = 0.25 (since, trials are independent)
P(TT) = P(T) P(T) = 0.5 * 0.5 = 0.25 (since, trials are independent)
P(HH) = P(H) P(H) = 0.5 * 0.5 = 0.25 (since, trials are independent)
P(TH) = P(T) P(H) = 0.5 * 0.5 = 0.25 (since, trials are independent)


Therefore,
  

It is clear that,
Xi ~ Ber (0.25) iid , i = 1(2)9
Xi ~ Ber (0.25) iid , i = 2(2)8

Now, it is known that,



It is also known that, for a random variable A ~ Bin(n,p) E(A) = np
Therefore,
E(Y) = 5 * 0.25 =1.25
Also,
E(Z) = 4 * 0.25 = 1

Now, according to the problem,
X = Y + Z






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