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4) A joint probability density function is given f(x, y) = (x + 2y) for 0 < x < 1, 0 < y < 1 and elsewhere. (a) Find the marginal probability density function h(y) and obtain conditional probability ensity function f(x = x1Y = y). (b) Find P(XSIY = 3).

4) A joint probability density function is given f(x, y) = (x + 2y) for 0 < x < 1, 0 < y < 1 and elsewhere. (a) Find the marginal probability density function h(y) and obtain conditional probability ensity function f(x = x1Y = y). (b) Find P(XSIY = 3).

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter12: Probability
Section12.CR: Chapter 12 Review
Problem 83CR
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1) A teacher is asked to select a male or a female student from one of the two N (Noisy) or S (Silent) classes at random. In N, there are 2 female and 3 male students and in class S, there are 3 male and 6 female students. Suppose that the teacher goes to the noisy class (Event N) with probability 2/3. (a) Find the probability that the teacher selects a female student (Event F). (b) Find the probability that the teacher goes to the noisy class first and select a female student. And then show that the events N and F are not independent. (c) Assume, we are given the information that a female student is selected. Find the conditional probability that the teacher had gone to the noisy class. 2) There are 5 Red, 3 Green and 2 Black balls in a box. Two balls are selected and removed in succession from the box. (a) List the elements of the sample space, corresponding probabilities and corresponding values w of the random variable W, where W is the number of Red balls selected. (b) Find the distribution function of the random variable W and plot its graph. 3) There are 2 Blue, 4 Red and 6 Yellow balls in a box. Two balls are selected at random from a box containing 2 Blue, 4 Red and 6 Yellow balls. Consider random variables X and Y as, respectively, the numbers of Blue and Red balls. (a) Find the probabilities associated with all possible pairs of values of X and Y, obtain the joint probability distribution function f(x, y). Draw a probability table corresponding to the values of X, Y. (b) Find the joint cumulative distribution function F(1,1). (c) Find the marginal distributions g(x) and h(y) for each x, y within the range of X and Y. Redraw the table with marginal values. (d) Find the conditional distribution of X given Y = 1. 4) A joint probability density function is given f(x, y) = (x + 2y) for 0 < x < 1, 0 < y < 1 and 0 elsewhere. (a) Find the marginal probability density function h(y) and obtain conditional probability density function f(x = xy = y). (b) Find P(X ≤1Y=2).
Transcribed Image Text:1) A teacher is asked to select a male or a female student from one of the two N (Noisy) or S (Silent) classes at random. In N, there are 2 female and 3 male students and in class S, there are 3 male and 6 female students. Suppose that the teacher goes to the noisy class (Event N) with probability 2/3. (a) Find the probability that the teacher selects a female student (Event F). (b) Find the probability that the teacher goes to the noisy class first and select a female student. And then show that the events N and F are not independent. (c) Assume, we are given the information that a female student is selected. Find the conditional probability that the teacher had gone to the noisy class. 2) There are 5 Red, 3 Green and 2 Black balls in a box. Two balls are selected and removed in succession from the box. (a) List the elements of the sample space, corresponding probabilities and corresponding values w of the random variable W, where W is the number of Red balls selected. (b) Find the distribution function of the random variable W and plot its graph. 3) There are 2 Blue, 4 Red and 6 Yellow balls in a box. Two balls are selected at random from a box containing 2 Blue, 4 Red and 6 Yellow balls. Consider random variables X and Y as, respectively, the numbers of Blue and Red balls. (a) Find the probabilities associated with all possible pairs of values of X and Y, obtain the joint probability distribution function f(x, y). Draw a probability table corresponding to the values of X, Y. (b) Find the joint cumulative distribution function F(1,1). (c) Find the marginal distributions g(x) and h(y) for each x, y within the range of X and Y. Redraw the table with marginal values. (d) Find the conditional distribution of X given Y = 1. 4) A joint probability density function is given f(x, y) = (x + 2y) for 0 < x < 1, 0 < y < 1 and 0 elsewhere. (a) Find the marginal probability density function h(y) and obtain conditional probability density function f(x = xy = y). (b) Find P(X ≤1Y=2).
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