1. Suppose X is a random variable with moment generating function,

M(t) = 2
3 e6t + 1
6 e12t + 1
6 e30t for all t:

(a) Is X continuous or discrete? Justify our assertion.
(b) If X is continuous, then compute its density function. If X is discrete, then compute its mass function.

2. Suppose X takes the values 1, 2, and 5 with respective probabilities 2=3, 1=6, and 1=6. Compute the moment generating function of 6X.

3. Two fair dice are rolled independently. Calculate Var(X1X2) where Xi denotes the number of dots rolled on the ith roll for i = 1; 2.

4. Let R be the collection of all points (x ; y) in the plane such that: x 0; y 0; and 4 x2 + y2 9:
Compute PfY > 2Xg when (X ; Y ) is drawn uniformly at random from the shaded region R.

5. Let R be the collection of all points (x ; y) in the plane such that: x 0; y 0; and 4 x2 + y2 9: Are X and Y independent? Prove or disprove.

6. Let X be a random variable with cumulative distribution function,
F(x) =
8><
>:
1 if x 1;
x2 if 0 x < 1;
0 if x < 0:

(a) Is X continuous or discrete? Justify your answer.

(b) If your answer to (a) was \continuous," then compute the density function of X. Otherwise, compute the mass function of X.

(c) What are:

i. PfX 0:5g;
ii. Pf0:2 < X 1:5g;
iii. Pf0:2 X < 1:5g?

7. Let X be a random variable with cumulative distribution function,
F(x) =
8><
>:
1 if x 1;
1=2 if 0 x < 1;
0 if x < 0:
(a) Is X continuous or discrete? Justify your answer.

(b) If you answered (a) as \continuous," then compute the density func- tion of X. Else, compute the mass function of X.

(c) What are:

i. PfX 0:5g;
ii. Pf0:2 < X 1g;
iii. Pf0:2 X < 1g?

8. Suppose that the random point (X ; Y ) has joint density,
f(x ; y) =
(
3
2 max(x ; y) if 0 < x < 1 and 0 < y < 1;
0 otherwise:
(a) Are X and Y independent? Prove or disprove.
(b) Compute PfX > 1=2g.

9. Let X be a random variable with density function,
f(x) =
8< :
ex
e  1
if 0 x < 1;
0 otherwise:

(a) Compute E(X) and Var(X). Show the details of your computation.

(b) Compute the moment generating function of X.

10. Consider a random variable X that satis es
PfX = xg =
(
C2 x if x = 2; 3; : : :;
0 otherwise;
where C is a nite and positive constant.

(a) Compute C. You may use, without proof the following fact from your calculus course: 1 + r + + rn = (rn+1 1)=(r  1) for all r > 0 and integers n 0.

(b) Calculate the cumulative distribution function of X.

(c) Prove that Pf140 < X 213g = 2a 2b where a and b are positive
integers, and compute a and b.

11. A certain basketball player makes n shot attempts. Her past history shows that she makes her shots 75% of the time, and misses 25% of the time. Assume that she shoots her n shots independently.

(a) Suppose n = 4. Calculate the probability that she makes at least 90% of her shots, if n = 4.

(b) Suppose n = 400. Use the central limit theorem to approximate the probability that she makes at least 90% of her shots.

12. Suppose X and Y are two independent random variables, both following an exponential distribution with parameter = 1.

(a) What is the joint density function of (X ; Y )?
(b) What is the density function of Z = max(X ; Y )?
(c) What is the expected value of Z := max(X ; Y )?
(d) What is the variance of Z := max(X ; Y )?

13. Compute E(XY ), where the random point (X ; Y ) has joint density, f(x ; y) = ( 3 2 max(x ; y) if 0 < x < 1 and 0 < y < 1; 0 otherwise:

14. Let X and Y denote two independent random variables, each following a standard normal distribution. De ne two new random variables,
U := X + Y p 2 ; V := X  Y p 2 :

(a) Compute the joint density of U and V .
(b) Compute the marginal densities of U and V .
(c) Are U and V independent? Prove or disprove.

15. Let X and Y denote two independent random variables, each following a uniform distribution on (0 ; 1). De ne two new random variables,
U :=
X + Y
p
2
; V :=
X  Y
p
2
:
Are U and V independent? Prove or disprove.

16. Suppose we pick a random point (X ; Y ) uniformly on the circle of radius one about the origin (0 ; 0) of the plane. De ne two new random variables,
U :=
X + Y
p
2
; V :=
X  Y
p
2
:
Compute the joint density of (U ; V ).
3

17. Suppose X and Y are independent with respective densities fX and fY .

(a) Prove that the density of Z := X + Y is
fZ(a) =
Z 1
1
fX(x)fY (a x) dx for all a:
(b) Suppose X and Y are independent and have the common density
fX(b) = fY (b) =
(
eb if b > 0;
0 otherwise:
Then use the previous part to compute the density of Z := X + Y .

18. Two fair dice are rolled independently. Let Di denote the number of dots rolled by die number i, where i = 1; 2, and then set X1 := max(D1 ;D2) and X2 := D1 + D2.

(a) Find the joint probability mass function of (X1 ;X2).
(b) Find PfX2 < 1:5X1g.

19. Let X and Y be independent random variables with respective mass functions fX and fY .

(a) Prove that the mass function of Z := X + Y is
fZ(a) =
X
x
fX(x)fY (a  x) for all a:

(b) Use part (a) to prove that if X has a Poisson distribution with pa- rameter 1 and Y has a Poisson distribution with parameter 2, and if X and Y are independent, then X + Y has a Poisson distribution with parameter 1 + 2.

20. Prove that if X and Y are two independent random variables, both dis- tributed uniformly on (0 ; 1), then X + Y is not uniformly distributed. [Hint. Examine the behavior of the function MX+Y (t) := E[et(X+Y )] as t ! 1.] 4 5