1. (10%) Solve the differential equation ⎧⎪⎪
⎨⎪⎪⎩
y′+ (tan t) y = t sin 2t, −π2 < t <π2, y(0) = 1.
2. (15%) Solve the differential equation dxdy = eysin x.
3. (10%) A chemical reaction invloves the collision of one molecule of a substance P with one molecule of a substance Q to produce one molecule of a new substance X. This is P+Q→X. If p, q, (p ≠ q) are the initial concentrations of P and Q, respectively, and let x(t) be the concentration of X at time t. Then p − x (t) and q − x (t) are the concentrations of P and Q at time t, respectively. The reaction is given by the equation
dx
dt = α (p − x) (q − x) , where α is a positive constant. If x(0) = 0, solve x (t) .
4. (10%) Let X be a random variable, taking values in{1, 2}. Let E(X) =53. (a) (5%) Find P(X = 1) and P(X = 2).
(b) (5%) Evaluate Var(X).
5. (10%) Evaluate∫−∞∞ e−x2+2bx+cdx, b, c∈ R. (You can use ∫−∞∞e−x2dx=√π.)
6. (10%) Let X, Y be random variables such that fX(t) = e−t, t≥ 0 and Y = 2X + 1. Find fY(t).
7. (10%) Let X, Y be two independent random variables both with the probability density fX(t) = fY (t) = √1πe−t2, t∈ R. Find the probability density function fW(t) of the random variable W = (X + Y )2.
8. (10%) The average number of phone calls is 20 per hour. Find the probability for the event of at least one phone call within an interval of 3 minutes. (Assume the validity of Poisson process for this case.)
9. (15%) Let X be the random variable for the duration of telephone calls within a certain city with the probability density function
fX(t) =2
5e−2t5, t> 0 where t denotes the duration in minutes of a randomly selected call.
(a) (4%) What percentage of calls last one minute or less ?
(b) (4%) What percentage of calls last between one and two minutes ? (c) (4%) What percentage of calls last 3 minutes or more ?
(d) (3%) What is the average length of a call ?
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