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2.1 Solve the below problem (show all calculations). Let Y, and Y, be random variables with joint density function f,y) - { Os y, s1, Osy; s1, 0sy; +y, S1 0, 2, elsewhere. The marginal density function of f1) = 2(1-y,), 0sy, s1 Identify the distribution of the marginal density function f(y,) and use it to find the E(Y;) and V(Y;).

2.1 Solve the below problem (show all calculations). Let Y, and Y, be random variables with joint density function f,y) - { Os y, s1, Osy; s1, 0sy; +y, S1 0, 2, elsewhere. The marginal density function of f1) = 2(1-y,), 0sy, s1 Identify the distribution of the marginal density function f(y,) and use it to find the E(Y;) and V(Y;).

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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Question 2 2.1 Solve the below problem (show all calculations). Let Y, and Y, be random variables with joint density function Os y, s1, Osy, S 1, 0sy, +y, <1 elsewhere. 2, ro,y) = { 0, The marginal density function of f(yı) = 2(1– y,), 0s y, s1 Identify the distribution of the marginal density function f(yi) and use it to find the E(Y;) and V(Y,). 2.2 Solve the below problem. Table 1 contains the probabilities associated with each possible pair of values for Y,and Y2 and is known as the joint probability function for Y,and Y2 Table: Probability function for Y, and Y2 y1 yz 01 2 01/9 2/9 1/9 1 2/9 2/9 0 2 1/9 0 0 Find F(1,-2) and F(3,3). 2.3 BP petrol is to be stocked in a bulk tank once at the beginning of each week and then sold to individual customers. Consider the joint density of Y,, the proportion of the capacity of the tank that is stocked at the beginning of the week and Y2, the proportion of the capacity sold during the week, given by 0 y2 s y1 S 1, elsewhere. 3y, 3/4 In each week an average of 70% of the petrol is sold and an average of 30% of the petrol remains in the Caltex bulk tank at the end of each week. In any given week of the year, what is the probability that the amount of petrol sold during the week is more than the amount of petrol that was stocked in the beginning of the week? 2.4 Complete the below statement. If the covariance of two random variables is zero, the variables 2.5 Complete the below statement. If two random variables are the joint probability can be written as the product of the marginal probabilities. 2.6 Solve the below problem: Let Y, and Y, have joint density function Se-Oi+y2), fV1, Y2) = 0, Yı 2 0, y2 2 0, elsewhere. 2.6.1 Find the marginal density function for Y,. Identify this density. (3.5) 2.6.2 For any y, > 0, what is the conditional density function of Y, given that Y2 = y2? (2.5) :::
Transcribed Image Text:Question 2 2.1 Solve the below problem (show all calculations). Let Y, and Y, be random variables with joint density function Os y, s1, Osy, S 1, 0sy, +y, <1 elsewhere. 2, ro,y) = { 0, The marginal density function of f(yı) = 2(1– y,), 0s y, s1 Identify the distribution of the marginal density function f(yi) and use it to find the E(Y;) and V(Y,). 2.2 Solve the below problem. Table 1 contains the probabilities associated with each possible pair of values for Y,and Y2 and is known as the joint probability function for Y,and Y2 Table: Probability function for Y, and Y2 y1 yz 01 2 01/9 2/9 1/9 1 2/9 2/9 0 2 1/9 0 0 Find F(1,-2) and F(3,3). 2.3 BP petrol is to be stocked in a bulk tank once at the beginning of each week and then sold to individual customers. Consider the joint density of Y,, the proportion of the capacity of the tank that is stocked at the beginning of the week and Y2, the proportion of the capacity sold during the week, given by 0 y2 s y1 S 1, elsewhere. 3y, 3/4 In each week an average of 70% of the petrol is sold and an average of 30% of the petrol remains in the Caltex bulk tank at the end of each week. In any given week of the year, what is the probability that the amount of petrol sold during the week is more than the amount of petrol that was stocked in the beginning of the week? 2.4 Complete the below statement. If the covariance of two random variables is zero, the variables 2.5 Complete the below statement. If two random variables are the joint probability can be written as the product of the marginal probabilities. 2.6 Solve the below problem: Let Y, and Y, have joint density function Se-Oi+y2), fV1, Y2) = 0, Yı 2 0, y2 2 0, elsewhere. 2.6.1 Find the marginal density function for Y,. Identify this density. (3.5) 2.6.2 For any y, > 0, what is the conditional density function of Y, given that Y2 = y2? (2.5) :::
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