MATH-UA 233/ MA-UY 3014

Theory of Probability/ Applied Probability

Summer 2024

Final: Sample Problems

The final will likely have 6-8 questions. The questions marked with ∗ are considered more difficult than the rest.

1. Find n, m ∈ N for which

2. Find n ∈ N, n ≥ 119, for which

3. For n ∈ N and x, y, z ∈ R, z > 0, find a closed formula for

Hint: Show the product of the first two factors is a binomial coefficient.

4. Suppose x, y > 0, with

Find x, y assuming that x − y = 6.

5. Suppose X is a random variable with E[X2 ] < ∞. Find a ∈ R minimizing

E[(X − 2a) 2 + (1 − aX) 2 ].

That is, find a such that for all b ∈ R,

E[(X − 2b) 2 + (1 − bX) 2 ] ≥ E[(X − 2a) 2 + (1 − aX) 2 ].

6. Find r ∈ R for which there exists a probability P on the sample space S = Z satisfying

P({n}) = 10 · r 3n ,                   P({−n}) = 2−n−4 ,

for all n ∈ N.

7. Independent events A, B, C satisfy P(A) = 2/1, P(A ∪ B ∪ C) = 18/13, P(B) + 2P(C) = 108/29. Find P(A ∩ B ∩ C c ).

8. A roulette is spun 300 times: its outcomes (the spins are independent) are integers between 1 and 400 (inclusively) and are denoted by Sk, 1 ≤ k ≤ 300. Let E be the event that S2k−1 + S2k = 100 + k for 1 ≤ k ≤ 100, and F the event that S3k−1 − S3k = 100 + k for 1 ≤ k ≤ 100. Which event has larger probability, E or F?

9(∗). Suppose X, Y are independent random variables with X ≥ 0, X  Y, and E[X] = 1. Show

E[e XY ] ≥ E[e X].

10. Suppose X has moment generating function

Find the moment generating function of X2 (a power series decomposition suffices).

11. A random variable X is called symmetric if X  −X.

(a) Suppose X is discrete: find the necessary and sufficient conditions for X to be symmetric in terms of its probability mass distribution.

(b) Find all discrete random variables X ∈ Z for which both X, X2 − 9 are symmetric.

(c) Show there is no random variable X (not necessarily discrete) such that X2 − 4X + 5 is symmetric.

12. Suppose Y = X3 + 4, and X is an exponential random variable with parameter 2 : i.e.,

p(x) = 2e −2x ,    x > 0

is the pdf of X. Compute the joint cumulative distribution of X and Y.

13(∗). Suppose X and Y = X3 − 4X are independent with E[X] = 0, E[X4 ] < ∞. Show E[X2 ] ≤ 4, and E[X4 ] ≤ 16.

14. Suppose X > 0, Y = log X, and

Find V ar(X), and Corr(X, X2 ).

15(∗). Suppose X takes finitely many values, and

Find the probability mass distribution of X.

16. Suppose X > 0, and

Show P(X ≤ 100 + ϵ) = 1 for any fixed ϵ > 0, while P(X ≥ 12) < 1.

17. Find an, bn ∈ R such that

18. Give an example of a sequence Xn with V ar(Xn) = 1, E[Xn] = 0, |Corr(Xi , Xj )| ≤ 2/1 for all i ≠ j, for which

does not hold.

Hint: Consider Xk = αkX + Yk for carefully chosen X, αk, Yk.

19(∗). Let Xn be a sequence of independent random variables with P(Xn = 1) = P(Xn = n) = 2/1. Prove

does not hold.

Hint: For a given N, compute P(Xk < k, ∀k ≥ N).

20. Suppose X, Y have joint probability density

Show X and Y are not independent.

21. Suppose f, g : R → R are nondecreasing functions, and X, Y are independent real-valued random variables with the same cumulative distribution function.

(a) Show

(f(X) − f(Y ))(g(X) − g(Y )) ≥ 0.

(b) Show

Cov(f(X), g(X)) ≥ 0.

22. Suppose X, Y, Z are random variables and

V ar(X) = V ar(Y ) = V ar(Z) = 1.

(a) By using V ar(aX + bY + cZ) ≥ 0, show

6Cov(X, Y ) + 15Cov(Y, Z) + 10Cov(Z, X) ≥ −19.

(b) Assume equality occurs in (a). Compute

Corr(6X + 3Y + 5Z, 3X).

23. Suppose X is a nonnegative random variable with

E[Xk ] ≤ 2 k k 3

for all k ∈ N, and fix ϵ > 0.

(a) Show that for any increasing function h : [0, ∞) → R and a > 0,

P(X ≥ a) = P(h(X) ≥ h(a)).

(b) Using Markov’s inequality and part (a), show

P(X < 2 + ϵ) = 1.

24. Suppose X1, X2, ... , Xn are random variables with

Cov(Xi , Xj + Xi) = 3i + j,          1 ≤ i ≤ j ≤ n.

(a) Find for all 1 ≤ i ≤ j ≤ n,

Cov(Xi , Xj ).

(b) Suppose X1, X2, ... , Xn have mean 1. Show that for any a > 0,

25. The joint density of X and Y is given by

(a) Compute the joint moment generating function of X and Y.

(b) Compute the individual moment generating functions of X and Y.

26. The moment generating functions of X and Y are

Suppose X, Y are independent. Find

E[(X + Y ) 2 ].

27. Suppose the moment generating function of X is

MX(t) = e (t+1)3−2t+8−a .

(a) Find a.

(b) Suppose Y is independent from X, and X + Y is a normal random variable with mean 0 and variance 2. Find the moment generating function of Y.

28. Suppose X1, X2, ... , Xn are i.i.d. random variables with probability density function

Find some a, b ∈ R for which

29. Suppose X1, X2, ... , Xn, ... are random variables with mean 0, and

for all i, j ≥ 1. Prove

30. A fair die is rolled 100 times and Ri is the value of the top face of the i th roll. Compute an approximation of

P(R1R2...R100 ≥ a 100)

for a > 1 using the central limit theorem.

31. Let Rn be the top face of the i th roll of a fair die. For n ≥ 100, Let

Xn = R1R2...R100 + R2R3...R101 + ... + Rn−99Rn−98...Rn.

(a) Find E[Xn].

(b) Show

Hint: Chebyshev’s inequality.

32. Find a, b ∈ R for which there is a random variable X with probability density function f given by

f(x) = ax + b, x ∈ [0, 2],

and

33. Find a, b ∈ R for which there exist probability density functions f, g given by

f(x) = ax + b, x ∈ [0, 2],

g(x) = 2bx + 2a, x ∈ [0, 1].

34. Suppose X has probability density function

Find a ∈ R with

25. Suppose X, Y are continuous random variables with probability densities f, g, and

Find

E[3Y + 8].

36. Suppose X, Y have joint probability density function

f(x, y) = x + y, 0 ≤ x, y ≤ 1.

Find E[XY ].

37. Suppose X, Y have joint probability density function

f(x, y) = cye−x + e −2x , 0 ≤ y ≤ 1, x > 0.

Find fX, fY , the marginal probability densities of X, Y.

38(∗). Suppose X is a continuous random variable with probability density

f(x) = e −a(x−t) 2 , x ∈ R.

Find t ∈ R that minimizes V ar((t + 1)X + 1) + V ar(tX) i.e., compute t0 ∈ R such that

V ar((s + 1)Y + 1) + V ar(sY ) ≥ V ar((t0 + 1)X + 1) + V ar(t0X)

for all s ∈ R with Y having probability density

g(x) = e −a(x−s) 2 , x ∈ R.

39. Suppose X, Y have joint probability density function

Show X, Y are not independent.

40. Suppose X, Y are independent random variables with variance 1. Find a ∈ R such that

41. Show there are no independent random variables X, Y, Z with

E[XY ] = 10, E[Y Z2 ] + Cov(X, 2Z + 3) = −2, E[Z 2X] + Cov(X + 2Y, 2Z + 3) = 10.

42. Suppose X, Y are jointly continuous random variables with joint probability density function

f(x, y) = e −x−2y + e −y/4−8x , x, y > 0.

Find a with fX|Y =a(0) = 2.

43. Suppose X, Y are independent, and so are X, X + Y. Show E[X2 ] = (E[X])2 , and find

Cor(X, X2 + 4X − 2).

44. Suppose X, Y have Cov(X, 2Y ) = 10, V ar(X) = 2, V ar(X + Y ) = 18. Find Corr(X − 10, 2Y + 4.)

45. Find a ∈ R such that the random variable X with moment generating function

has E[X2 ] = 5.

46. Suppose Xn are i.i.d. with X1  X, E[X] = 1, E[X2 ] = 2. Show

47. Suppose X, Y are random variables with

Show X, Y are normal random variables, and find their means and variances.

48. Suppose Xn are i.i.d. with X1  N(0, 1), and let an be a sequence of real numbers with P n≥1 a 2 n = 6. Find c ∈ R with

c(a1X1 + ... + anXn) ⇒ N(0, 54).

Hint: The sums above are normal random variables.

49. Suppose X, Y are random variables with

Find c ∈ R such that

for Xn i.i.d., and Yn i.i.d. with X1  X, Y1  Y.

50. Suppose Xn are i.i.d. with X1  X, V ar(X) = 2, E[X4 ] = 100. Find a > 0 such that

51. Suppose X is a standard normal random variable and Y = a + bX + cX2 . Show

52. Suppose X, Y are independent uniform. random variables on [0, 1], [3, 4], respectively.

(a) Find

Cov(X + Y, X − 5).

(b) Find a, b ∈ R satisfying