Consider an experiment in which you toss a fair coin (i.e., the probability of t
ID: 3219436 • Letter: C
Question
Consider an experiment in which you toss a fair coin (i.e., the probability of turning up heads is 1/2 for each toss) n times. All the tosses are independent. Answer the following questions:
(I) Let Xi be a 01 random variable which is 1 if the total number of heads you saw in tosses 1,2,...,i is even. The random variable Xi is defined for each index i = 1,2,...,n. Let X = X1 + X2 + ... + Xn. What is the expected value E[X]?
(II) Give the best bound you can, in terms of n, for the following tail probability Pr[|X E[X]| E[X]] where 0 < 1.
(III) Let random variables Y1, . . . , Yn be defined as follows: Yi = 1 if the number of heads you have seen till the ith toss is divisible by 3. Are Y1, . . . , Yn independent?
Explanation / Answer
1) Xi = 1 if total no of heads in i tosses is even. And 0 if number of heads is odd.
X1 =0 always as only one head can come
X2 = 1 if two heads come with probability 1/4 or 0 if one or no head comes
EX = 0+ 1/4 + 1/4 + (1/4+1/16) + (1/4 + 1/16) + (1/4+ 1/16+1/64) +.....
= 2*1/4 + 2*(1/4+1/16) + 2*(1/4 +1/16+1/64)+....
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