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Continuous Random Variables - Probablity - Exam, Exams of Probability and Statistics

Alliance UniversityProbability and Statistics

This is the Exam of Probablity which includes Cumulative Distribution Function, Random Variable, Probability Density Function, Triangle, Uniformly Distributed, Complement, Density, Marginal Densities, Independent etc. Key important points are: Continuous Random Variables, Marginal Density Function, Density Functions, Convolution Formula, Uniform, Interval, Marginal Density, Joint Density Function, Conditional Density, Different Schools

Typology: Exams

2012/2013

Uploaded on 02/20/2013

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MA 587/687 (Advanced Probability), Dr. Chernov Final Exam
10 problems. April, 26, 2012
587 students: do 8 problems for full credit. (If you do more, you get extra points.)
687 students: do 9 problems for full credit. (If you do more, you get extra points.)
1. Let Xand Ybe continuous random variables with joint density function
fXY (x, y ) = ½cy for x≤y≤√x, 0≤x≤1,
0 otherwise
(a) Find the value of the constant c.
(b) Find the marginal density function of X.
(c) Find the marginal density function of Y.
2. Let Xand Ybe independent random variables with density functions
fX(x) = ½2xfor 0 ≤x≤1,
0 otherwise
fY(y) = ½3y2for 0 ≤y≤1,
0 otherwise
Find the density function for X+Yby using the convolution formula.
3. The conditional distribution of X, given Y, is uniform on the interval [Y, 2Y]. The
marginal density of Yis
fY(y) = ½3y2for 0 ≤y≤1,
0 otherwise
(a) Find the joint density function fXY (x, y).
(b) Find the conditional density of Y, given X=x > 0.
4. In a math tournament, 30 teams from different schools participate. Each team consist
of four members. All the 120 participants are to be seated in 10 rooms, with 12 par-
ticipants in each room. Suppose the seating is done at random. We say that a team is
lucky if all its members end up seating in the same room. What is the expected number
of lucky teams in this tournament?
5. Independent random variables X,Yand Zare identically distributed. Let W=X+Y.
The moment generating function of Wis MW(t) = (0.2+0.8e−t)8.
(a) Find the moment generating function of V=X+Y+Z.
(Bonus) Find possible values of Xand the respective probabilities.
pf3

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MA 587/687 (Advanced Probability), Dr. Chernov Final Exam 10 problems. April, 26, 2012 587 students: do 8 problems for full credit. (If you do more, you get extra points.) 687 students: do 9 problems for full credit. (If you do more, you get extra points.)

  1. Let X and Y be continuous random variables with joint density function

fXY (x, y) =

cy for x ≤ y ≤

x, 0 ≤ x ≤ 1 , 0 otherwise

(a) Find the value of the constant c. (b) Find the marginal density function of X. (c) Find the marginal density function of Y.

  1. Let X and Y be independent random variables with density functions

fX (x) =

2 x for 0 ≤ x ≤ 1 , 0 otherwise

fY (y) =

3 y^2 for 0 ≤ y ≤ 1 , 0 otherwise

Find the density function for X + Y by using the convolution formula.

  1. The conditional distribution of X, given Y , is uniform on the interval [Y, 2 Y ]. The marginal density of Y is

fY (y) =

3 y^2 for 0 ≤ y ≤ 1 , 0 otherwise

(a) Find the joint density function fXY (x, y). (b) Find the conditional density of Y , given X = x > 0.

  1. In a math tournament, 30 teams from different schools participate. Each team consist of four members. All the 120 participants are to be seated in 10 rooms, with 12 par- ticipants in each room. Suppose the seating is done at random. We say that a team is lucky if all its members end up seating in the same room. What is the expected number of lucky teams in this tournament?
  2. Independent random variables X, Y and Z are identically distributed. Let W = X+Y. The moment generating function of W is MW (t) = (0.2 + 0. 8 e−t)^8. (a) Find the moment generating function of V = X + Y + Z. (Bonus) Find possible values of X and the respective probabilities.
  1. Let X 1 ,... , Xn be independent and identically distributed random variables, each of them is exponential with a common parameter λ > 0. Let Yn be the minimum of X 1 ,... , Xn, i.e., Yn = min{X 1 ,... , Xn}.. (a) Find the cumulative distribution of Yn. (b) Show that Yn converges in probability to 0 by showing that for arbitrary ε > 0

lim n→∞ P(|Yn − 0 | ≤ ε) = 1.

(Bonus) Show that the same is true when X 1 ,... , Xn are independent and identically distributed random variables that are nonnegative and have a common density function f (x) such that f (x) > 0 for all 0 < x < 1.

  1. Let X and Y be the number of hours that a randomly selected person watches movies and sporting events, respectively, during a three-month period. The following information is known about X and Y :

E(X) = 30 E(Y ) = 20 σX = 5 σY = 4 ρX,Y = 0. 2

Four hundred people are randomly selected and observed for these three months. Let T be the total number of hours that these four hundred people watch movies or sporting events during this three-month period. Approximate the value of P(T > 21000).

1

2 3 4

5 6

  1. A rat runs through the maze shown in the above figure. At each step it leaves the room it is in by choosing at random one of the doors out of the room. (a) Give the transition matrix P for this Markov chain. (b) Is it an irreducible (i.e., ergodic) chain? (c) Is it a regular (i.e., irreducible and aperiodic) chain? (d) Find the fixed (i.e., stationary) probability vector.