Solve the following problem:
Q1: I have 100 items of a product in stock. The probability mass function for the product's demand D is P(D=90)=P(D=100)=P(D=110)=1/3
a) find the mass function, mean and variance of the number of items sold.
b) find the mass function, mean, and variance of the amount of demand that will be unfilled because of the lack of stock.
Q2: Let X be the following discrete random variable: P (X = -1) =P (X =0) = P (X=1) = 1/3. Let Y= X2. Show that COV (X, Y) =0 but X and Y are not independent random variables.
Q3: I draw five cards from a deck replacing each card immediately after it is drawn. I receive four dollars for each heart that is drawn. Find the mean and variance of my total payoff.
Q4: Let X be a continuous random variable with density function:
4 - x (x) = k , 0 x < 4
a ) what is k?
b) Find the CDF for X
c) Find E(X) and Var(X)
d) find E(X) and the variance of X.
Q5: Suppose two dice are tossed
a) what is the probability that the total of the two dice add up to 7 or 11 ?
b) what Is the probability that the total of the two dice will add up to a number other than 2 or 12?
c) are the events
d) Are the events
E1 = first die shows a three and E2= total of the two dice is seven independent events
e)given that the total of the two dice is five what is the probability that the first die shows two
f) given that the first die shows a five what is the probability that the total of the two dice is even