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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 5-2.1 Joint Probability Distribution Definition Computing Joint Probabilities b d Pr a X 1 b, c X 2 d f xy ( x, y ) dy dx a c 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: f xy ( x, y ) c, 8.5 x 10.5, 120 y 240 10.5 240 10.5 8.5 120 8.5 c dy dx 1 or c 240 120 dx 1 120c 10.5 8.5 1 240c 1 or c 1/ 240 .0041667 find c! Example: Computing Joint Probabilities Thinking outside the cuboid A cuboid is a solid figure bounded by six rectangular faces: a rectangular box Now find: Pr 9 X 10 and 125 Y 140 10 140 10 1 1 15 dy dx .0625 140 125 dx 240 240 9 240 9 125 c 1 240 9 10 125 x 140 y Volume = (1) (140-125) / 240 = .0625 Example 5-12 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 5-2.2 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 1 f xy ( x, y ) , 8.5 x 10.5, 120 y 240 240 240 1 240 120 1 f x ( x) dy , 8.5 x 10.5 240 240 2 120 10.5 f y ( y) 8.5 1 2 1 dx , 120 y 240 240 240 120 Pr 150 Y 200 1 200 150 dy .41667 150 120 120 200 Example 5-13 Example 5-13 5-2 Two Continuous Random Variables 5-2.3 Conditional Probability Distributions Definition 5-2.3 Conditional Probability Distributions Example 5-14 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 5-2.4 Independence Definition Is our blood example independent? 1 1 1 f xy ( x, y ) f x ( x) f y ( y ) 240 2 120 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. 12 f xy ( x, y ) x 2 x y , 0 x 1, 0 y 1 5 1 1 12 12 xy 2 2 0 0 5 x 2 x y dy dx 5 0 2 xy x y 2 dx 0 1 1 1 1 12 x 12 2 x3 x 2 12 12 4 3 2 2 x x dx x 1 5 0 2 5 3 4 0 5 12 students matriculating Marginal Distribution of X 1 1 12 12 x f x ( x) x 2 x y dy 2 x y dy 5 5 0 0 1 12 x y 12 x 2 y xy 1.5 x 5 2 0 5 2 12 x 2 3 12 3 2 3 E[ X ] x dx x x dx 5 2 5 02 0 1 1 1 12 x x 12 1 3 .6 5 2 4 0 5 4 5 3 4 Marginal Distribution of Y 1 1 12 12 f y ( y ) x 2 x y dx 2 x x 2 xy dx 5 5 0 0 1 12 2 x x y 12 2 y x 5 3 2 0 5 3 2 3 2 1 12 y 2 y 12 2 y y 2 E[Y ] dy dy 5 3 2 5 0 3 2 0 1 1 y 12 1 2 12 y .4 6 0 5 6 5 5 3 2 3 The Variances 1 12 3 3 12 3x x 21 4 E[ X ] x x dx 5 02 5 8 5 0 50 21 2 x .62 .06; .245 50 1 4 5 2 1 12 2 y y y 12 2 1 7 12 2 y E[Y ] dy 5 0 3 2 8 0 5 9 8 30 5 9 7 2 y .42 .0733; y .271 30 1 2 2 3 3 4 The Covariance and Correlation 1 1 12 12 2 2 E[ XY ] xy x 2 x y dy dx x 2 y xy y dy dx 5 5 0 0 0 0 1 1 1 1 12 2 2 y y 12 12 x x x 1 x y x dx x 2 1 dx x 2 dx 5 2 3 0 5 0 2 3 5 0 2 3 0 0 1 2 3 1 1 3 4 x3 12 1 1 1 7 12 x x 5 3 8 9 0 5 3 8 9 30 7 Cov( XY ) xy (.4)(.6) .006667 30 xy .006667 xy .10 x y .245 .271 1 3 2 A Conditional Distribution 12 f xy ( x, y ) 5 x 2 x y 2 x y fY | x ( y ) 3 12 3 f x ( x) x x x 2 5 2 Given that 50 percent of the students that submit applications visit the campus, what is the expected number that will not attend? 3 y 3 fY | x .5 ( y ) 2 y 3 1 2 2 2 1 3 y 2 y3 9 4 3 E (Y | x .5) y y dy .4167 3 0 12 2 4 0 1 E (Y | x .8) .3814 Example 5-18 5-4 Bivariate Normal Distribution Definition Figure 5-17. 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 X1, X2, …,Xp are independent normal RV with E{xi] = i and V[Xi] = i2, then p Y ci X i is normal i 1 p p i 1 i 1 with E[Y ] ci i and V [Y ] ci2 i2 Gosh, everything is so normal. The Last Example Let Xi = 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 Y p c X i 1 Item 1 2 3 4 5 6 7 i i is the daily sales revenue Mean Variance Selling Price 8.5 1.2 $22.50 3.4 2.3 $18.75 2.1 0.87 $5.67 7.8 2.4 $24.50 11.2 3.2 $15.99 8.4 5.9 $36.80 15.1 6.7 $12.95 Expected revenue Variance 191.25 607.50 63.75 808.59 11.907 27.97 191.1 1,440.60 179.088 818.18 309.12 7,990.02 195.545 1,123.61 1141.76 12816.46 Std dev. $113.21 Pr{Y 1100} Y y 1100 1141.76 Pr 113.21 y Pr{z .37} .3561 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. Can you read me the story again about the marginal and conditional distributions. I liked that one the best! … and thus our story ends with the X random variable having a strong positive correlation with the Y random variable.