Question

Suppose Alice’s pocket expenses per month are normally distributed with mean 1000 dollars and standard deviation 100 dollars. Suppose Bob’s pocket expenses per month are normally distributed with mean 600 dollars and standard deviation 50 dollars. Assume that Alice’s and Bob’s pocket expenses are independent.

(a) Find the probability that Alice and Bob total pocket expense in one month exceeds 2000 dollars

(b) Find the probability that Alice spends twice as much as Bob in pocket expenses in some random month.

Answer 1

a) let X and Y are pocket expense in one month for Alice and Bob

A =X+Y

expected value of A =E(X)+E(Y) 1000+600 =1600

and standard deviation of A =sqrt(Var(X)+Var(Y)))=sqrt(100^2+50^2)=111.803

probability that Alice and Bob total pocket expense in one month exceeds 2000 dollars:

probability =P(X>2000)=P(Z>(2000-1600)/111.803)=P(Z>3.58)=1-P(Z<3.58)=1-0.9998=0.0002

b)

let B=X-2Y

expected value of B =1000-2*600 = -200

and standard deviation =sqrt(100^2+(2*50)^2)= 141.421

probability that Alice spends twice as much as Bob in pocket expenses in some random month:

probability =P(B>0)=P(Z>(0--200)/141.421)=P(Z>1.41)=1-P(Z<1.41)=1-0.9207=0.0793