1. Consider a discrete random variable X that takes on values 1, 2, 3, and 4 with probabilities p(1), p(2), p(3), and p(4). Let its mean be μ and standard deviation be σ.
a. Let b be a constant (some fixed number). Show that var(X + b) = var(X). You have to show this directly. [Hint: Consider the new random variable Y = X + b. Use its pmf to compute its variance.]
b. Let c be a constant. Show that E(c X) = c μ and var(cX) = c2 var(X). [Hint: Use the same
approach as in part (a)]
c. Let Y = X - μ, the centered version of X. Use your result from part (a) to determine var(Y).
d. Let Z = X/σ. Use your results from part (b) to determine the expected value and variance of Z.
2. The distribution of piston diameters is normally distributed. It is known that 10% of all pistons have diameters exceeding 20 inches and 5% have diameters smaller than 10 inches. What are the mean and SD of the distribution?
3. The distribution of resistance for certain types of resistors (measured in ohms) is normally distributed with mean μ and standard deviation σ. What should the value of σ be if we want to make sure than 90% of the resistors have resistances that are within 4 ohms of the mean μ?
4. A store has express and regular checkout lanes. Let X be the number of customers at the express lane on a randomly selected Monday at noon, and le Y be the number of customers at the regular lane. The joint pmf of X and Y are given below:
a) What is the probability that there is at least one customer in both lanes?
b) What is P(X ≤ 1 and Y ≤ 1)?
c) Describe the following event in words:
fX 6=0 and Y 6=0g
d) Are X and Y independent? Explain.
5. An automobile repair shop checks each car that comes in for repair to see if the tire pressure needs adjustment and if the headlights need to be properly focused. Let X be the number of tires that need pressure adjustment and Y denote the number of headlights that need to be focused for the next customer.
a. Suppose X and Y are independent with marginal pmf's:
pX(0) = 0.1, pX(1) = 0.2, pX(2) = 0.4, pX(3) = 0.2, and pX(4) = 0.1; pY(0) = 0.5, pY(1) = 0.3, and pY(2) = 0.2.
Write down the joint pmf of (X, Y).
b. Compute P(X > 3 and Y >1).
c. What is P(X + Y > 1)?
6. Let U and V be iid with mean μ and standard deviation σ.
a. What is var( ¼ U + ¾ V)?
b. What is var( ½ U + ½ V)?
c. Which has bigger variance?
7. Let X and Y be two independent discrete random variables. X is uniform on the integers {0, 1, 2, 3}. Y is binary, takes on values 1 and 2, and pY(y = 1) = 0.4.
a. Compute E(XY) and show that it equals E(X) E(Y).
b. Use this result to compute cov(X, Y).
8. Show the following:
a. Cov(X+5, Y+3) = cov(X, Y)
b. Cov(aX, bY ) = ab Cov(X, Y) where a and b are constants (fixed numbers).
c. What is the relationship between cov(X, Y) and cov(X, - Y)?
d. What is the relationship between cor(X, Y) and cor(X, - Y)?
e. If Cor(X, Y) = 0.5, what is Cor(2X, 5 - Y)?
9. If Y = 5X + 3, what is cor(X, Y)?
