Question

A fair coin will be tossed 200,000 times. Let X denote the number of Tails. (a) What is the expected value and the standard deviation of X? (b) Consider a game in which you have to pay $5 in order to earn $log10(X) when X > 0. Is this a fair game? If not, your expected profit is positive or negative?

Answer 1

Here X is random variable taking values 1,2,3... so on till we toss the coin 200000 times.

a) Please paste following code in R to get Expected value and SD.

x=seq(from=1, to=200000, by=1)
n=length(x)
n

expect=sum(x*(x/n))
exp

var= sum((x-m)^2*(x/n))
SD=sqrt(var)
SD

Which gives us Expected Value :  13333433334 and Standard Deviation:  4.21637e+12

b)

To pay $5 in order to earn $log10(X) when X > 0 is not a fair game since probability of having Number of tails in experiment is high which results into negative Profit.