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Review for Final Exam – Probability | MATH 3338, Exams of Probability and Statistics

Material Type: Exam; Professor: Chen; Class: Probability; Subject: (Mathematics); University: University of Houston; Term: Unknown 1989;

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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Math 3338 Review for Final Exam
Chapter 1: Only general concepts, such as, mean, median, variance,
sample, are needed. No graphing problems.
Chapter 2: Axioms of probability, independence of events, counting
techniques (permutations, combinations). No conditional probability.
Problem 1: Events A, B are independent. P(A) = .4, P (B) = .7.
Compute P(AB0).
Problem 2: We select 3 distinct letters from A,B,C,D,E,F and 2 distinct
numbers from 1,2,3,4,5 to make a password.
(a) How many different passwords can we make?
(b) Find the probability that a password contains letter A.
Chapter 3 and 4: Basic concepts such as discrete and continuous ran-
dom variables, pmf, pdf, cdf. Properties of pmf, pdf and cdf. Com-
putation of expected values, variances, moment generating functions.
Binomial distributions. Normal distributions. Transformation of ran-
dom variables.
Problem 3: Two six-sided fair dice are tossed independently. Let Xbe
the maximum of the two numbers of the tossed dice.
(a) Find the probability mass function p(x) of X.
(b) Compute the expected value of X.
Problem 4: Let MX(t) = e3t+t2and Y= 3X+ 2. Find E(Y), V (Y).
Problem 5: Let the pdf of Xbe
f(x) = (ax, 0x2;
0,otherwise.
(a) Determine the value of a.
(b) Compute P(0.5< X 1.5).
(c) Find cdf of X.
(d) Compute E(X) and V(X).
Problem 6: Assume that pdf of Xis f(x) = 2e2x, x 0; f(x) = 0
otherwise. Find MX(t). Use MX(t) to compute E(X), V (X).
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Math 3338 Review for Final Exam

  • Chapter 1: Only general concepts, such as, mean, median, variance, sample, are needed. No graphing problems.
  • Chapter 2: Axioms of probability, independence of events, counting techniques (permutations, combinations). No conditional probability. Problem 1: Events A, B are independent. P (A) =. 4 , P (B) = .7. Compute P (A ∩ B′). Problem 2: We select 3 distinct letters from A,B,C,D,E,F and 2 distinct numbers from 1,2,3,4,5 to make a password. (a) How many different passwords can we make? (b) Find the probability that a password contains letter A.
  • Chapter 3 and 4: Basic concepts such as discrete and continuous ran- dom variables, pmf, pdf, cdf. Properties of pmf, pdf and cdf. Com- putation of expected values, variances, moment generating functions. Binomial distributions. Normal distributions. Transformation of ran- dom variables. Problem 3: Two six-sided fair dice are tossed independently. Let X be the maximum of the two numbers of the tossed dice. (a) Find the probability mass function p(x) of X. (b) Compute the expected value of X. Problem 4: Let MX (t) = e^3 t+t

2 and Y = 3X + 2. Find E(Y ), V (Y ). Problem 5: Let the pdf of X be

f (x) =

{ ax, 0 ≤ x ≤ 2; 0 , otherwise.

(a) Determine the value of a. (b) Compute P (0. 5 < X ≤ 1 .5). (c) Find cdf of X. (d) Compute E(X) and V (X). Problem 6: Assume that pdf of X is f (x) = 2e−^2 x, x ≥ 0; f (x) = 0 otherwise. Find MX (t). Use MX (t) to compute E(X), V (X).

Problem 7: In a city, 10% of the population do not have medical in- surance. A random sample of 400 people is selected. What is the probability that the number of uninsured people is between 30 and 70 (inclusive). Problem 8: Suppose X has uniform distribution on [0, 1] and pdf of Y is fY (y) = 3e−^3 y^ , y ≥ 0; fY (y) = 0, otherwise. Is there a map g such that Y = g(X)?

  • Chapter 5: Concepts of joint pmf and pdf. Computation of expected values. Problem 9: X, Y are discrete random variables taking values in {0,1}. We know some values of joint pmf p(x, y): p(0, 0) = 0. 3 , p(1, 0) =
    1. 1 , p(1, 1) = 0.4. (a) Calculate p(0, 1). (b) Compute E(X(Y + 2)). Problem 10: Suppose X, Y are two independent continuous random variables. The pdf of X is fX (x) = 2x, 0 ≤ x ≤ 1; otherwise, fX (x) = 0. The pdf of Y is fY (y) = 38 y^2 , 0 ≤ x ≤ 2; otherwise, fY (y) = 0. (a) Please find the joint pdf of X, Y. (b) Compute E(XY + 3).
  • Chapter 6: Properties of X, T¯ o of a random sample, especially for nor- mal distributions. Use central limit theorem to do estimates. Problem 11: The weight of residents in a building has the normal distribution with μ = 160, σ = 15. The elevator’s capacity is 1000 lb’s. Compute the probability that the elevator is over its capacity when 6 people enter it. Problem 12: A random sample {X 1 , X 2 , · · · , X 100 } are i.i.d. with μ = 12 , σ = 5. Use central limit theorem to estimate P ( X¯ ≤ 12 .3).