MATH 4510: Review for Final Exam
1. Let Ube a uniform random variable on the interval (1,3). Let X= ln(U). Find the probability
density function of Xand compute the probability P(X≤1).
We compute FX(x) = P(X≤x) = P(lnU≤x) = P(U≤ex) = ex−1
3−1for 0 ≤x≤ln 3.
So, the probability density function of Xis
f(x) = d
dxF(x) = (ex
20<x<ln 3
0 otherwise .
So, P(X≤1) = R1
0f(x)dx=ex
2
1
0=e−1
2.
2. The random variables Xand Yare jointly continuous with joint probability density function
f(x, y) = cxy for 0 < x, y < 1 and f(x, y ) = 0 otherwise. Find the constant cand compute the
marginal probability density functions fXand fY. Find the conditional probability density
function of Xgiven that Y=y. Find P(X Y ≤0.5).
We have that 1 = R1
0R1
0cxy dxdy=c
4, so that c= 4. We compute:
fX(x) = Z1
0
4xydy= 2xfor 0 <x<1
fY(y) = Z1
0
4xydx= 2yfor 0 < y < 1, and
fX|Y(x|y) = f(x, y )
fY(y)=4xy
2y= 2xfor 0 <x<1.
Finally,
P(XY ≤0.5) = Z Zxy ≤0.5
4xy dA=Z0.5
0Z1
0
4xy dydx+Z1
0.5Z1
2x
0
4xy dydx
=Z0.5
0
2xdx+Z1
0.5
2x
4x2dx=x2
0.5
0+ln x
2
1
0.5=1
4+ln 2
2.
Aside: We see that f(x,y ) = fX(x)fY(y), so that Xand Yare independent, so we could have
instead computed fX|Yby noting that fX|Y(x|y) = fX(x) when Xand Yare independent.
3. A coin that lands heads with probability pis flipped ntimes. Let Xbe the number of heads
and Ybe the number of tails. Compute the correlation coefficient ρ(X, Y ).
We have that Xand Yare both binomial with parameters nand p, so if we let σ2=np(1−p),
then var(X) = var(Y) = σ2. Also, X+Y=n, so
cov(X, Y ) = cov(X, n −X) = cov(X, n)−cov(X, X) = −var(X) = −σ2.
Then,
ρ(X, Y ) = cov(X, Y )
pvar(X) var(Y)=−σ2
σ2=−1.
