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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
0
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 X
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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