4. (20 points) Let X ~ N(3,4)
(a) (6 points) Find variance, mean and standardize X
(b) (6 points) Express P(3 < X < 5) in terms of σ, and use 68-95-99.7 rule to estimate the probability
(c) (4 points) Express P(X > 7) in terms of σ, and use 68-95-99.7 rule to estimate the probability
(d) (4 points) Express P(X < -3) in terms of σ, and use 68-95-99.7 rule to estimate the probability
5. (15 points) Let X be an exponential random variable with mean 3. Find
(a) (3 points) E(X)
(b) (2 points) Var(X)
(c) (4 points) P(X > 3)
(d) (6 points) P(t < X < t + 15|X > t) for some t > 0
6. (10 points) Suppose X1, X2 and X3 be independent exponential random variables with mean 10.
(a) (3 points) Find the distribution of min{X1,X2,X3}
(b) (3 points) Find P(X1 < X2 < X3)
(c) (2 points) Find P(Xi < min{X2,X3})
(d) (2 points) Find P(min{X2,X3} < Xi < max{X2,X3})
7. (10 points) Suppose the joint PMF, p(x,y) of discrete random variables X and Y is shown in the table below:
p(x,y) y=0 y=1 P(x)
0=x .15 .1
x=1 .05 .2
x=2 .2 .3
P(y)
(a) (4 points) Find the marginal PMF P(x) and P(y)
(b) (2 points) Find P(y = 0|X = 1)
(c) (2 points) Find P(x = 2|Y = 1)
(d) (2 points) Determine whether X and Y are independent