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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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1) Joint Mass Function of (X, Y):

Given:

$X \sim Poisson (λ)$ where $λ$ is the mean number of passengers getting on the bus.
Each passenger gets off independently with probability $Error converting from MathML to accessible text.$ .

The joint mass function $P (X = x, Y = y)$ represents the probability that there are $Error converting from MathML to accessible text.$ passengers getting on the bus and $Error converting from MathML to accessible text.$ passengers getting off the bus.

For a given $x$ , the number of passengers getting off the bus ( $Y$ ) follows a binomial distribution with parameters $Error converting from MathML to accessible text.$ (number of trials, i.e., the number of passengers getting on) and $p$ (probability of success, i.e., the probability a passenger gets off):

$open parentheses table row x row y end table close parentheses p to the power of y left parenthesis 1 minus p right parenthesis to the power of y$

The Poisson distribution gives the probability of $Error converting from MathML to accessible text.$ passengers getting on the bus:

$e x p left parenthesis negative lambda right parenthesis space lambda to the power of x divided by x factorial$

Therefore, the joint mass function $P (X = x, Y = y)$ is the product of these two probabilities:

$left square bracket e x p left parenthesis negative lambda right parenthesis space lambda to the power of x divided by x factorial space right square bracket space space open parentheses table row x row y end table close parentheses p to the power of y left parenthesis 1 minus p right parenthesis to the power of y$

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