Question

Your friend is trying to figure out how much money he will spend at a food truck over the next 400 days. He buys all his coffee and falafel sandwiches there. Each coffee costs $1 and the number of cups of coffee he buys on day i is a random variable Ci with a Poisson distribution with mean = 1. Every day he also buys a $1 falafel sandwich at the food cart with a probability of 50%. The amount of coffee he buys each day is independent of whether he buys lunch at the cart, and his consumption of coffee and lunch is independent across days. (a) Let F be the amount he spends on falafel sandwiches over the next 400 days. Find E[F] and V(F).

Answer 1

Answer:

E(F)=200

V(F)=100

Step-by-step explanation:

(a) Let F be the amount he spends on falafel sandwiches over the next 400 days. Find E[F] and V(F).

If we analized his expenses in falafel sandwiches, we know he buys a $1 sandwich with aprobability of 50%.

If we call the random variable D the daily expense in falafel sandwiches, it is a Bernoulli variable with p=0.50.

If F is the sum of the daily expenses D during the 400 days, then we can say it is a binomial variable with p=0.5 and n=400.

Then we can calculate the expected value of F as:

$E(F)=nE(D)=400*0.5=200$

The variance of F is then

$V(F)=n*V(D)=n*p(1-p)=400*0.5*(1-0.5)=400*0.25=100$