1. (12%) Solve the differential equation (x2+ 1)dydx+ 4xy = x with the initial condition y(2) = 1.
2. (12%) (a) Solve the differential equation dydt = λy(y − 1), 0 < y < 1 and λ > 0 is a constant with the initial condition y(0) =12.
(b) Evaluate lim
t→∞y(t).
3. (12%) Evaluate∫−∞∞e−x2+4xdx. (You can use∫−∞∞e−x2dx=√ π.)
4. (12%) In a Poisson process, P(k, t) = (λt)k!ke−λt indicates the probability of k occurrences of a specific event in the time interval[0, t]. Let W denote the time of the second occurrence (counted from the beginning of the process).
(a) Find P(W > t).
(b) Find the probability density function fW(t) of W.
5. (16%) Let X be a random variable with the probability density function f(x) =xA2, 1≤ x ≤ e, where A is a constant.
(a) (6%) Find A.
(b) (5%) Find E(X).
(c) (5%) Find Var(X).
6. (16%) Rolling a fair dice, we define two random variables
X= {1 if the outcome is even
0 if the outcome is odd , Y = {1 if the outcome is in{1, 2, 3}, 0 if the outcome is in{4, 5, 6}.
Let Z= X + Y
(a) (6%) Are X and Y independent?
(b) (5%) Find E(Z).
(c) (5%) Find Var(Z).
7. (10%) Let X, Y be two independent random variables both with the probability density f(t) = λe−λt, t≥ 0. Find the probability density function fZ(t) of the random variable Z = X + Y .
8. (10%) Let X be the random variable with the probability density function fX(t) = √1πe−t2. Find the probability density function fW(t) of the random variable W = X2.
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