Answered: Let the joint pdf for the continuous…

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

Let the joint pdf for the continuous random variables X and Y be:

f(x,y) = { 6x² ; 0 < x <1, (1/3)x < y <x

0; elsewhere]

(i) Find P(X=0.6, Y=0.6)

(ii) Find P(X=0.6, Y<0.6)

(iii) What is the marginal pdf for X?

(iv) What is the marginal pdf for Y?

(v) Find P(Y > 0.8 | X = 0.9)

Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.

Expert Solution

Step 1: Solving with explanation

(i) To find P(X=0.6, Y=0.6), you need to calculate the probability that both X and Y equal 0.6. Since X and Y are continuous random variables, the probability of them having exact values is zero. So, P(X=0.6, Y=0.6) = 0.

ii>

>To find

P (X = 0.6, Y < 0.6)

, we integrate the joint pdf over the allowed range of

y

$integral subscript 0.6 divided by 3 end subscript superscript 0.6 end superscript 6 x squared space d y$

$= 6x 2 \cdot (0.6 - 0.2)$

$= 2.4 x 2$

$= 2.4 (0.6) 2$

$= 0.864$

(iii) To find the marginal pdf for $Error converting from MathML to accessible text.$ , you need to integrate the joint pdf over the range of $Error converting from MathML to accessible text.$ :

$integral subscript x divided by 3 end subscript superscript x 6 x squared space d y$

$= 6 x 2 \cdot (x - 1/3 \cdot x)$

$= 6 x 2 \cdot (2/3) x$

$= 4 x 3$

This is the marginal pdf for $Error converting from MathML to accessible text.$ .

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Let the joint pdf for the continuous random variables X and Y be: f(x,y) = { 6x2 ; 0 < x <1, (1/3)x < y 0.8 | X = 0.9)