1. (10%) Solve the differential equation y

1. (10%) Solve the differential equation y^′= y(y²− 1), where y is not a constant function.

2. (10%) Find the particular solution of xdy

dx = 2y + x³ln x, x> 0, satisfying y(1) = −1.

3. (15%) Assume the range IX of a random variable X is{−1, 0, 1}. Let E(X) = 0 and Var(X) =2 3. Find (a) (4%+4%) P(X = 1) and P(X = 0)

(b) (7%) Var(X²).

4. (10%) If X_i∼ X for all i = 1, 2, ⋯, 10 and {X¹, X₂,⋯, X¹⁰} is a set of independent random variable. Assume E(X) = 1 and Var(X) = 2. Find

(a) (5%) E(5X¹⋅ X²⋅ ⋯ ⋅ X¹⁰) (b) (5%) Var(X₁+ X²+ X¹⁰

10 )

5. (10%) Compute the integral: ∫

∞

−∞(2x + 1)e^−x2+6xdx.(You can use the result∫

∞

−∞

e^−x2dx=√ π.)

6. (15%) Let X, Y be two independent random variables with density function f_X(t) = f^Y(t) =⎧⎪⎪

⎨⎪⎪⎩

te^−t, t> 0 0, t≤ 0.. (a) (8%) Let Z= X + Y . Find f^Z(t).

(b) (7%) Let W = X². Find fW(t).

7. (15%) (5%+5%+5%) Let X be a random variable with density function fX(t) = 3e^−3t. Find (a) P(1 ≤ X ≤ 2), (b) E(X) and (c) Var(X).

8. (15%) Calls arrive to a cell in a certain wireless communication system according to a Poisson process with arrival rate 120 calls per hour. Find the following probabilities:

(a) (7%) The first call arrives after time t= 3.5.

(b) (8%) Two or fewer calls arrive in the first 3.5 minutes.

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