Question

1. 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 we have X: number of tails occurs in 200000 times coin tossed.

a) to obtain the E(X) and SD(X) run the following code in R

x=seq(from=1, to=200000, by=1)
n=length(x)
n
expect=sum(x*(x/n))
expect
var= sum((x-m)^2*(x/n))
SD=sqrt(var)
SD

We get the following values

E(X)=  13333433334

SD(X)= 4.21637e+12

b) Since we in order to earn the Log10(X) we have to pay $5 , which results into paying high amount so this will lead to negative profit. Hence not a fair game