Please show all work. Justify your numerical answer with brief explanations when

Question

Please show all work. Justify your numerical answer with brief explanations whenever possible. Conditional expectation (Ross 7.5) 1. (Applications of conditional expectation.) In the first part of the problem, you are askcability of heads is p. Let N be the numl>er of trials needed to obtain the first heads. Show that E[N] = l/p by conditioning upon the outcome of the first trial: E[N] = E[N]H on first trial]P(H on first trial) + E|N|T on first trial]. Find E|N|H on first trial) and E|N|T on first trial], plug them into the equation al>ove, and then solve for E[N]. (b) Suppose you flip a fair coin repeatedly, and stop when you obtain two heads in a row. How many times do you ex|>ect to Hip lieforv you stop? IIint: Condition u|mui the outcomes of the first two trials.|

Explanation / Answer

(a)

Either the event occurs on the first trial with probability p, or with probability 1 p the expected wait is 1 + E.
Therefore E = p·1 + (1 p)(1 + E), from which E = 1/p

Therefore the expected wait for a single head to appear = 1 / (1/2) = 2.

(b)

We now consider the expected wait for n > 1 consecutive heads. Let En be the expected wait for n consecutive heads.

In order to obtain n consecutive heads, we must first obtain n 1 consecutive heads, followed by:

It follows that En = (1/2)(En-1+ 1) + (1/2)(En-1 + 1 + En),

from which En = 2En-1 + 2.

So, for n=2,

E=2*2+2=6

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