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(a) Let Z be a discrete random variable taking one of the four distributions covered in Chapter 10. Suppose you know that = (k + 1)E(Z) where k is the last non-zero digit of your student ID number. Determine the distribution of Z and find its parameter(s), explaining your argument carefully.

(a) Let Z be a discrete random variable taking one of the four distributions covered in Chapter 10. Suppose you know that = (k + 1)E(Z) where k is the last non-zero digit of your student ID number. Determine the distribution of Z and find its parameter(s), explaining your argument carefully.

A First Course in Probability (10th Edition)
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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(a) Let Z be a discrete random variable taking one of the four distributions covered in Chapter 10. Suppose you know that Var(Z) = (k + 1)E(Z) where k is the last non-zero digit of your student ID number. Determine the distribution of Z and find its parameter(s), explaining your argument carefully.
Transcribed Image Text:(a) Let Z be a discrete random variable taking one of the four distributions covered in Chapter 10. Suppose you know that Var(Z) = (k + 1)E(Z) where k is the last non-zero digit of your student ID number. Determine the distribution of Z and find its parameter(s), explaining your argument carefully.
Erample 11.1: Coloured balls A bag contains three red balls, two yellow balls, and two green balls. Suppose that we pick three balls at random (without replacement). Let R denote the number of red balls we pick, and Y the nunber of yellow balls we pick. Find P(R= 1, Y = 1) and P(R = 3,Y = 1). Solution: We can treat this situation as unordered sampling without replacement (see Section 3.4). As we are picking three balls from a set of seven balls, we have, S 3D 3 The event (R= 1)n(Y 1) is the event that we pick one red ball (from three red balls), one yellow ball (from two yellow balls), and one green ball (from two green balls). Since all outcomes are equally likely, we can calculate the probability of this event as: (R = 1)n(Y = 1)_ G)) 3× 2 x 2 12 %3D P(R 1,Y = 1) = 35 35 It is even easier to calculate P(R 3,Y 1); it is impossible to draw three red balls and one yellow ballif we only draw three balls in total so (R=3)n(Y%3D1)%3= 0 and P(R 3,Y 1)=0. Erercise 11.2: Coloured balls (revisited) In the set-up of Example 11.1, evaluate P(R=r,Y = y) for all possible values of R and Y. Use your results to complete the following table for the joint p.m.f.: 3 12/35 The next proposition relates the joint distribution P(X-,Y y) and the So-called marginals P(X ) and P(Y y)- Proposition 11.2. Let X and Y be two discrete random variables defined on the same sample space and taking values r1,22,... and y1,2.... respectively. The marginal distribution of X can be obtained from the joint distribution as P(X- ) -FOx - . Y = y).
Transcribed Image Text:Erample 11.1: Coloured balls A bag contains three red balls, two yellow balls, and two green balls. Suppose that we pick three balls at random (without replacement). Let R denote the number of red balls we pick, and Y the nunber of yellow balls we pick. Find P(R= 1, Y = 1) and P(R = 3,Y = 1). Solution: We can treat this situation as unordered sampling without replacement (see Section 3.4). As we are picking three balls from a set of seven balls, we have, S 3D 3 The event (R= 1)n(Y 1) is the event that we pick one red ball (from three red balls), one yellow ball (from two yellow balls), and one green ball (from two green balls). Since all outcomes are equally likely, we can calculate the probability of this event as: (R = 1)n(Y = 1)_ G)) 3× 2 x 2 12 %3D P(R 1,Y = 1) = 35 35 It is even easier to calculate P(R 3,Y 1); it is impossible to draw three red balls and one yellow ballif we only draw three balls in total so (R=3)n(Y%3D1)%3= 0 and P(R 3,Y 1)=0. Erercise 11.2: Coloured balls (revisited) In the set-up of Example 11.1, evaluate P(R=r,Y = y) for all possible values of R and Y. Use your results to complete the following table for the joint p.m.f.: 3 12/35 The next proposition relates the joint distribution P(X-,Y y) and the So-called marginals P(X ) and P(Y y)- Proposition 11.2. Let X and Y be two discrete random variables defined on the same sample space and taking values r1,22,... and y1,2.... respectively. The marginal distribution of X can be obtained from the joint distribution as P(X- ) -FOx - . Y = y).
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