### Random variable with moment generating function

Assignment Help Basic Statistics
##### Reference no: EM13919464

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:

(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:

(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

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