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Q1 (10 points)

f(x) g(y) ∫ f(x) dx = g(y) dy = 1

−∞

If and are non-negative functions such that , let the pair of continuously −∞

fX,Y (x, y) = f(x)g(y)

distributed random variables have p.d.f. given by .

a) Find the marginal distributions of and ;

X Y

b) Find the conditional distribution of given that ; and the conditional distribution of given that . X Y = y Y X = x

c) Explain why the answers in parts a) and b) are unsurprising.

Q2 (10 points)

X Y {1, 2, 3, 4, 5} fX,Y (x, y)

Suppose and are discrete random variables taking values in where is proportional to  x × y fX,Y (x, y) = cxy 1 ≤ x, y ≤ 10 c

, that is for . Find the value of (Hint: the distributive law might make the calculations simpler).

Q3 (10 points)

(X,Y )

The random variables are jointly continuously distributed and have density

fX,Y (x, y) = {e−y

a) Calculate the marginal p.d.f’s and ;

fX fY

b) Find the conditional distributions and . fX|Y (x|y) fY|X (y|x)

c) Are and independent?

X Y

if y ≥ x ≥ 0; otherwise.

d) Interpret the conditional distributions that you obtained in part b) above. (Are these conditional distributions distributions that you have seen before?)

Q4 (10 points)

(X,Y ) {(x, y):x, y ≥ 0;x + y ≤ 1} P(X > 2Y ) Let be uniform random variables on the region . Compute .

Q5 (10 points)

X ∼ N(μ, ) σ2

Let .

a) Write down the m.g.f., of ;

MX

t u(x) = etx P(X > z)

b) If is a fixed positive number, write , and use Markov’s inequality to estimate in terms of  MX (z)

c) If is fixed, find the best (i.e. minimum) estimate for that you can obtain by changing the number . z P(X > z) t

Q6 (10 points)

X λ fX (x) = λe−λx x ≥ 0 Let be an exponential random variable with parameter (that is, for ). Y ⌊X⌋ Z Z = frac(X) = X − Y

Let be the discrete random variable and be the continuous random variable , the X

integer and fractional parts of respectively.

a) Find the c.d.f. of (it may help to think about which values of give rise to values of that are ); Y X Y ≤ n

b) Find the c.d.f. of (it may help to think about which values of give rise to values of that are ; you may need Z X Z ≤ t

to sum a geometric series);

(Y ,Z) a(1 + r + + … + ) r

c) Find the joint c.d.f. of . (Hint: the sum of a finite geometric series, is

2 rn

a(1 − )/(1 − r r)

n+1

.)

d) Show that and are independent.

Y Z

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