1. (12%) Evaluate the improper integral ∫
∞ 0+
e−3x
√ 2xdx.
2. (12%) If electricity power failures occur according to a Poisson distribution with an average of 7 failures every 15 days, calculate the probability that there will be more than one failure during a particular day.
3. (13%) Let X, Y be independent random variables with probability density functions given by fX(t) = te−t, fY(t) =1
2t2e−tfor t ≥ 0,
and fX(t) = fY(t) = 0 for t < 0. Find the probability density function fZ(t), where Z = X + Y .
4. (15%) Let X and Y be two independent random variables. X and Y both take value at {−1, 1}. Suppose we know their joint probability as follows:
P (X = 1, Y = −1) = 8 15 P (X = −1, Y = −1) =
4 15 P (X = 1, Y = 1) =
2 15 P (X = −1, Y = 1) =
1 15 Let Z =X ⋅ Y + 1
2 .
(a) (3%) Find the range of Z and the prabability function of Z.
(b) (3% each, 6% total) Find E(Z) and Var(Z)
(c) (2% each, 6% total) Suppose Wn∼Z and are independent with each other. Let W = W1+W2+ ⋅ ⋅ ⋅ +W10.
Find P(W = 4), expectation E(W ) and varance Var(W )
5. (12%) Let the random variable W denote the waiting time in a fast food restaurant. It is known that the average of W is 2 minutes and that W has the exponential distribution.
(a) (3%) Write down the probability density function of W . (b) (3%) Find the probability P(1 < W < 3).
(c) (3% each, 6% total) Find the expectation E(W ) and the variance Var(W ). Please write down your detail calculation.
6. (12%) Solve the differential equation (t2+2)y′(t) + (4t)y(t) = 1 satisfying y(0) = 2.
7. (12%) Solve the differential equation y′(t) = sin t + sin t(y(t))2. Find the general solution.
8. (12%) Let X be a continuous random variable with the probability density function fX(t) = 1
√πe−t2. We define a new random variable W = X3. Find the corresponding probability density of the random variable W .
Page 1 of 1
