Q9 Consider a sequence of Bernoulli trials with probability

Q9

Consider a sequence of Bernoulli trials with probability of \"success\" p. Suppose that in each trial you earn $1 if the outcome is \"success\", whereas you pay $1 if the outcome is \"failure\". Let Sn your profit after n trials. What is your expected profit after n trials? Suppose that n is odd and find P(Sn = 0). Is Sn a Binomial random variable?

Solution

let X be the random variable denoting the profit in each trial.

so X:     $1                 -$1

P[X=x]= p                  1-p

so E[X]=1*p-1(1-p)=2p-1

a) after n trials expected profit=E[S_n]=n*E[X]=2np-n   [as the trials are independent]   [answer]

b) P[Sn=0]=P[profit is zero]=P[number of success=number of failures]

so no. of success=no. of failures=n/2

but since n is odd n/2 cant be an integer

so P[Sn=0]=0 [answer]

c) Sn=X1+X2+...Xn   where Xi the value of X at ith trial

as sum of bernoulli events follows binomial distribution. so Sn follows a binomial distribution.