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Probability Distributions - Introductory Statistics - Lecture Notes, Study notes of Mathematical Statistics

Chhattisgarh Swami Vivekanand Technical UniversityMathematical Statistics

These are the important key points of lecture notes of Introductory Statistics are: Probability Distributions, Bivariate and Multivariate, Number of Contracts, Probability Function, Construction Jobs, Joint Probability Density Function, Random Variables, Proportions, Marginal and Conditional Probability Distributions, Marginal Density

Typology: Study notes

2012/2013

Uploaded on 01/11/2013

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Stat 366 Lab 3 Problems (September 26, 2006) page 1
TA: Yury Petrachenko, CAB 484, [email protected], http://www.ualberta.ca/∼yuryp/
Bivariate and Multivariate Probability Distributions
5.1 Contracts for two construction jobs are randomly assigned to one or more of three firms, A,
B, and C. Let Y1denote the number of contracts assigned to firm A, and Y2the number of
contracts assigned to firm B. Recall that each firm can receive 0, 1, or 2 contracts.
(a) Find the joint probability function for Y1and Y2.
(b) Find F(1,0).
5.6 Let Y1and Y2have the joint probability density function given by
f(y1, y2) =
k y1y2,0≤y1≤1,0≤y2≤1,
0,elsewhere.
(a) Find the value of kthat makes this a probability density function.
(b) Find the joint distribution function for Y1and Y2.
(c) Find P(Y1≤1/2, Y2≤3/4).
5.12 Suppose that the random variables Y1and Y2have joint probability density function f(y1, y2)
given by
f(y1, y2) =
6y2
1y2,0≤y1≤y2, y1+y2≤2,
0,elsewhere.
(a) Verify that this is a valid joint density function.
(b) What is the probability that Y1+Y2is less than 1?
5.14 Let Y1and Y2denote the proportions of time (out of one working day) during which employees
I and II, respectively, perform their assigned tasks. The joint relative frequency behavior of
Y1and Y2is modeled by the density function
f(y1, y2) =
y1+y2,0≤y1≤1,0≤y2≤1,
0,elsewhere.
pf2

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Stat 366 Lab 3 Problems (September 26, 2006) page 1 TA: Yury Petrachenko, CAB 484, [email protected], http://www.ualberta.ca/∼yuryp/ Bivariate and Multivariate Probability Distributions 5.1 Contracts for two construction jobs are randomly assigned to one or more of three firms, A, B, and C. Let Y 1 denote the number of contracts assigned to firm A, and Y 2 the number of contracts assigned to firm B. Recall that each firm can receive 0, 1, or 2 contracts. (a) Find the joint probability function for Y 1 and Y 2. (b) Find F (1, 0).

5.6 Let Y 1 and Y 2 have the joint probability density function given by

f (y 1 , y 2 ) =

k y 1 y 2 , 0 ≤ y 1 ≤ 1 , 0 ≤ y 2 ≤ 1 , 0 , elsewhere. (a) Find the value of k that makes this a probability density function. (b) Find the joint distribution function for Y 1 and Y 2. (c) Find P (Y 1 ≤ 1 / 2 , Y 2 ≤ 3 /4).

5.12 Suppose that the random variables Y 1 and Y 2 have joint probability density function f (y 1 , y 2 ) given by f (y 1 , y 2 ) =

6 y^21 y 2 , 0 ≤ y 1 ≤ y 2 , y 1 + y 2 ≤ 2 , 0 , elsewhere. (a) Verify that this is a valid joint density function. (b) What is the probability that Y 1 + Y 2 is less than 1?

5.14 Let Y 1 and Y 2 denote the proportions of time (out of one working day) during which employees I and II, respectively, perform their assigned tasks. The joint relative frequency behavior of Y 1 and Y 2 is modeled by the density function

f (y 1 , y 2 ) =

y 1 + y 2 , 0 ≤ y 1 ≤ 1 , 0 ≤ y 2 ≤ 1 , 0 , elsewhere.

Stat 366 Lab 3 Problems (September 26, 2006) page 2 (a) Find P (Y 1 < 1 / 2 , Y 2 > 1 /4). (b) Find P (Y 1 + Y 2 ≤ 1).

Marginal and Conditional Probability Distributions

5.17 Continuing Exercise 5.1.

(a) Find the marginal probability distribution of Y 1. (b) According to results in Chapter 4, Y 1 has a binomial distribution with n = 2 and p = 1/3. Is there any conflict between this result and the answer you provided in (a)?

5.22 Continuing Exercise 5.6.

(a) Find the marginal density functions for Y 1 and Y 2. (b) Find P (Y 1 ≤ 1 / 2 | Y 2 ≥ 3 /4). (c) Find the conditional density function of Y 1 given Y 2 = y 2. (d) Find the conditional density function of Y 2 given Y 1 = y 1. (e) Find P (Y 1 ≤ 3 / 4 | Y 2 = 1/2).

5.28 Continuing Exercise 5.12.

(a) Show that the marginal density of Y 1 is a beta density with α = 3 and β = 2. (b) Derive the marginal density of Y 2. (c) Derive the conditional density of Y 2 given Y 1 = y 1. (d) Find P (Y 2 < 1. 1 | Y 1 = .60).