345 ACTIVITY 9: Using Joint, Marginal, and Conditional Probability Distributions.

WHY:

This activity will help clarify the concept of a joint probability density function and examine how to use this function to determine marginal and conditional distributions. In this course, we will often be interested in the interaction of several random variables and, thus, we must know how to compute probability using joint distributions. This topic requires most of the base concepts we reviewed earlier in the course.

LEARNING OBJECTIVES:

1. Discover how to compute probability using the joint density function for a pair of continuous random variables.
2. Understand the relationship of marginal and conditional distributions to both the joint distribution and the probability distributions of the component random variables.
3. Gain confidence in the ability of the team to facilitate learning.

CRITERIA:

1. Quality of the answers to the Critical Thinking Questions.
   • Completeness
   • Clarity
   • Depth
2. Level of trust and participation in team learning.

INFORMATION:

Joint Probability Distribution f(x,y) of Discrete Random Variables X, Y

1. f(x, y) >= 0
2. The sum of all values of f(x,y) = 1
3. f(x, y) = P(X = x and Y = y)

Note that for any event A in the joint sample space, P((X,Y) is in A) is the sum of f(x,y) for all (x,y) in A

Joint Density Function f(x, y) for continuous Random Variables X and Y.

1. f(x, y) >= 0.
2. The double integral from -infinity to infinity of f(x,y) = 1
3. P((X,Y) is in A) = the double integral over A of f(x, y).

The Marginal Distributions g(x) with respect to X, and h(y) with respect to Y

g(x) = the integral over the range of Y of f(x, y) dy and h(y) = the integral over the rsange of X of f(x, y) dx for continuous Random Variables.

g(x) = the sum over the values of Y of f(x, y) and h(y) = the sum over the values of X of f(x,y) for discrete Random Variables.

Conditional Distribution

f(x | y) = f(x, y)/h(y) and f(y | x) = f(x, y)/g(x).

Independent Random Variables

X and Y are independent iff f(x,y) = g(x) * h(y)

RESOURCES:

1. Section 3.5: Probability and Statistics for E & S
2. Base Concepts
3. 30 minutes

PLAN:

1. Choose roles if you have not already done so.
2. Work through the model, verifying each step.
3. Answer the critical thinking questions.

MODEL:

Two electronic components of a phone switching system work in harmony for the success of the total system. Let X and Y denote the life expectancy in years of the two components. The joint density function of X and Y is given by

                              (3x - y)/9	1 <= x <= 3 and 1 <= y <= 2
                f(x, y)  =
     				  0		elsewhere

Find the probability that the sum of the life expectancies is at least 3. To answer this question it is helpful to draw the domain of f(x, y) and sketch the region that satisfies the event description.

			|
			|
			|____1____2____3__
		      1/   ____	   _ -
		      /   /   /_ -  x + y = 3
		    2/   /  _/
		    /   / - /
		  3/_ -/___/

From the diagram we can see that the following double integral will find the desired probability:

The double integral of (3x - y)/9 dxdy where x ranges from 3 - y to 3 and y ranges from 1 to 2

= 1/9 times the integral from 1 to 2 of 3x²/2 - yx evaluated from 3-y to 3

= 1/9 times the integral from 1 to 2 of 9y - 5y²/2 = 23/27

Now we can find the marginal density function, g(x), for X using the following computation:

g(x) = the integral from 1 to 2 of (3x - y)/9 dy = x/3 - 1/6 1<=x<=3

We can use the marginal density function to find the probability that X > 2:

P(X > 2) = the integral from 2 to 3 of x/3 - 1/6 = 2/3

Finally, we can compute the probability that Y is between 1 and 1.5 given that X is 2. We first compute the conditional distribution f(y|x) = f(x,y)/g(x) and then compute the conditional probability.

		(3x - y)       6x - y   12 - y
     f(y|x) = ------------  =  ------ = ------ when x = 2
	      9(x/3 - 1/6)     6x - 3      9

P(1 < Y<1.5 | X=2) = the integral from 1 to 1.5 of (12 - y)/9 = 19/36

CRITICAL THINKING QUESTIONS:

These will be provided in class.

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