SAMPLE PROBLEMS FOR FINAL EXAM
MATH 627 Probability, 12/12
1. Let X1, X2, X3be the outcomes of rolling a fair die three times. Deter-
mine P(X1+X2+X3≤9).
2. Let Xand Ybe random variables with the joint probability density
function given by
fXY (x, y ) = 3x0<y<x<1
0 otherwise
Compute E[X|Y=1
2].
3. Let X1have the binomial b(n1,1
2) distribution and let X2have the
binomial b(n2,1
2) distribution. Determine the distribution of Ywhere
Y=X1−X2+n2.
4. Let {X1, X2, ...., X25}be 25 independent Bernoulli random variables
each with the probabilities P(X= 1) = 1
2and P(X= 0) = 1
2. Let
Y= Σ25
i=1Xi. Use the Central Limit Theorem to approximate P(12 <
Y < 14).
5. Let Xbe a random variable that is uniformly distributed on (0,1). If
Y=sin(2πX) then find P(Y > 0).
6. Let Xbe an N(0,1) random variable and y=max(0, X). Compute
E[Y].
7. Two digits are randomly selected with replacement from the collection
{0,1, ..., 9}. Find the probability that their sum is 15.
8. Let Xand Ybe random variables with the joint probability density
function
fXY (x, y ) = 10x2y0<y<x<1
0 otherwise
Let V=X+Yand W=X2. Compute P(V < 1).
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