Chapter 5 Joint Continuous Probability Distributions Doubling our pleasure with two random variables Chapter 5C.

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Presentation transcript:

Chapter 5 Joint Continuous Probability Distributions Doubling our pleasure with two random variables Chapter 5C

Today in Prob/Stat

5-2 Two Continuous Random Variables Joint Probability Distribution Definition

Computing Joint Probabilities That’s a mighty fine formula you got there.

Example: Computing Joint Probabilities In a healthy individual age 20 to 29 yrs, the calcium level in the blood, X, is usually between 8.5 and 10.5 mg/dl and the cholesterol level is usually between 120 and 140 mg/dl. Assume for a healthy individual the random variables (X,Y) is uniformly distributed with a PDF given by: find c!

Example: Computing Joint Probabilities Now find: x y 9 10 Volume = (1) ( ) / 240 =.0625 Thinking outside the cuboid A cuboid is a solid figure bounded by six rectangular faces: a rectangular box

Example 5-12

Figure 5-8 The joint probability density function of X and Y is nonzero over the shaded region.

Example 5-12

Figure 5-9 Region of integration for the probability that X < 1000 and Y < 2000 is darkly shaded.

5-2 Two Continuous Random Variables Marginal Probability Distributions Definition

Returning to our blood example Let X = a continuous RV, the calcium level in the blood, Y = a continuous RV, the cholesterol level in the blood

Example 5-13

5-2 Two Continuous Random Variables Conditional Probability Distributions Definition

5-2.3 Conditional Probability Distributions

Example 5-14

Figure 5-11 The conditional probability density function for Y, given that x = 1500, is nonzero over the solid line.

Conditional Mean and Variance

5-2 Two Continuous Random Variables Independence Definition

Is our blood example independent?

A Dependent Example Let X = a continuous RV, the fraction of student applicants (deposits) that visit the University campus Let Y = a continuous RV, the fraction of student applicants (deposits) that do not attend the University. students matriculating

Marginal Distribution of X

Marginal Distribution of Y

The Variances

The Covariance and Correlation

A Conditional Distribution Given that 50 percent of the students that submit applications visit the campus, what is the expected number that will not attend?

Example 5-18

5-4 Bivariate Normal Distribution Definition

Figure Examples of bivariate normal distributions.

Example 5-30 Figure 5-18

Marginal Distributions of Bivariate Normal Random Variables

Figure 5-19 Marginal probability density functions of a bivariate normal distributions.

5-4 Bivariate Normal Distribution

Example 5-31

A Reproductive Continuous Distribution If X 1, X 2, …,X p are independent normal RV with E{x i ] =  i and V[X i ] =  i 2, then Gosh, everything is so normal.

The Last Example Let X i = a normal RV, the daily demand for item i stocked by the Mall-Mart Discount Store. Mall-Mart has a daily cost requirement of $1100. Will they have a cash flow problem? Then is the daily sales revenue

Other Reproductive Continuous Distributions The sum of independent exponential RVs having the same mean is Erlang (gamma) The sum of independent gamma RVs having the same scale parameter is gamma The sum of independent Weibull RVs having the same shape parameter is Weibull The overachieving student will want to research the derivations of these results.

So Ends Chapter 5 Another great adventure has just ended. … and thus our story ends with the X random variable having a strong positive correlation with the Y random variable. Can you read me the story again about the marginal and conditional distributions. I liked that one the best!