Lecture 14. Joint Probability Distribution. Joint Probability Distribution. Generally we must deal with two or more random variable simulatneously.

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1 ENM 07 Lecture 4 Joint Probability Distribution Joint Probability Distribution Generally we must deal with two or more random variable simulatneously. For example, we might select abricated vidget and measure height weight o it. Thus, both height and weight are the random variables o interest. The objective o this subject is to ormulate joint probability distribution or two or more random variables present methods or obtaining both marginal and conditional distributions. present a deinition o independence or random variables deine covariance and correlation

2 Joint Probability Distribution or Two R.Vs I the possible values o [x,x ] are either inite or countably ininite in number, then [x,x ] will be a two dimensional discrete random vector. I possible values o [x,x ] are some uncountable set in the Euclidean plane, then [x,x ] will be a two dimensional continuous random vector. Joint Probability Distribution or Two R.Vs Discrete Case (, ) (, ) (, ) 0 and p( x, x ) p x x p X x X x p x x all j all i

3 Joint Probability Distribution or Two R.Vs Example: x P(x) x 0 /30 /30 /5 /0 /30 4/5 /30 /30 /0 /5 3/0 /30 /5 /0 /5 3 /30 /0 /5 4 /0 /0 P(x) 7/30 7/30 4/5 7/30 /30 top p(x) this is the tabular representation. This means that x 0 and x 0 with probability /30. Joint Probability Distribution or Two R.Vs Graphical representation o a bivariate probability distribution

4 Joint Probability Distribution or Two R.Vs I [X,X ] is a continuous random vector with range space R in the Euclidean plane, then the joint density unction has the ollowing properties. ( x, x ) 0 or all ( x x ) R, ( x x ), dxdx R ( x x ), dx dx Joint Probability Distribution or Two R.Vs. Continuous Case P b b, a a ( a X b, a X b ) ( x x ) It should again be noted that doesn t represent the probability o anything, and the x convention, x that ( ) dx dx ( x, x ) 0 or ( x x ) R,

5 Joint Probability Distribution or Two R.Vs We can think o (x,y) as speciying a surace at height (x,y) above the point (x,y) in a three-dimensional coordinate system. (x,y) ( x, y) ] P[ A is the volume under this surace and above the region A. y a shaded rectangle x Joint Probability Distribution or Two R.Vs Example A bank operates both a drive-up acility and walk up window. On a randomly selected day, let xthe proportion o time that the drive up acility is in use ( at least one customer is being served or waiting to be served) ythe proportion o time that the walk up window is in use. Then the set o possible values or (x,y) is the rectangle {( x, y) : 0 x,0 } D y the joint p.d. o (x,y) is given by 6 ( ) ( x + y ) x, y x,0 y o. w.

6 Joint Probability Distribution or Two R.Vs a) Is this unction a p.d.? b) Find probability that neither acility is busy more than one quarter o time. Joint Probability Distribution or Two R.Vs Example A nut company markets cans o deluxe mixed nuts containing almonds, and peanuts. Suppose the net weight o each can is exactly pound, but the weight contribution o each type o nut is random. Let ; Xthe weight o almonds in a selected can Ythe weight o cashews Then the region o positive density is R {( x, y) : 0 x, 0 y, x + y } ( x, y) 4xy 0 0 x,0 y, x + y o. w

7 Joint Probability Distribution or Two R.Vs The shaded region pictured in igure is as ollows. y (0,) (x,-x) x (,0) x Show that this unction is p.d.. Joint Probability Distribution or Two R.Vs Example ( x, x ) x 0.5, 0 x 00 0 ow.. Compute ( 0. 0., 00 00) P x x

8 Marginal Distributions Having deined the bivariate probability distribution, called the the joint probability distribution ( or in the continuous case the joint density), a natural question arises as to the distribution o x or x alone. These distributions are called marginal disributions. In discrete case, the marginal distribution o X is; P ( x ) p x, x ) i,,... ( all j the marginal distribution o X is P ( x ) p( x x ) j,,3..., all i Marginal Distributions Graphical representation o marginal distribution or X P(X ) 8/30 9/30 6/30 4/30 3/ X Graphical representation o marginal distribution or X

9 Marginal Distributions In continuous case, the marginal distribution o X is ( ) x ( x, x ) dx the marginal distribution o X is ( ) x ( x, x ) dx Marginal Distributions Example 6 ( ) ( x + y ) x, y x,0 y o. w. a) Find marginal p.d.. o x, which gives the probability distribution o busy time or the drive-up acility without reerence to the walk-up window

10 Marginal Distributions c) Find the marginal p.d. o Y d)ind probability that 3 P y 4 4 Expected Value and Variance. Discrete Case ( ) µ ( ) (, ) E x x p x x p x x all i x p x x ( ) (, ) all i all j all i xp x ( x ) σ ( x ) p( x ) ( µ ) p( x x ) v µ all i all i all j x, x p( ) x p( x x ) x all i all j µ i j, µ

11 Expected Value and Variance ( x ) µ x p( x ) x p( x, x ) E j j i V ( x ) σ ( x µ ) p( ) ( µ ) p( x x ) x j j i x, x p( ) µ x p( x, x ) µ j i x Expected Value and Variance Example. The mean and variance o x could be determined using the marginal distribution o x. E V ( x ) ( x ) µ [ ] [ ] σ

12 Expected Value and Variance. Continuous Case E ( x ) µ x ( x ) dx x ( x, x ) dxdx ( x ) σ ( x ) ( x ) V µ dx ( ) ( ) x µ x, x dxdx x ( x ) dx µ x ( x x ) dx dx, µ Expected Value and Variance E V ( x ) µ x ( x ) dx x( x, x ) ( x ) σ ( x µ ) ( x ) ( x µ ) ( x, x ) x x ( x ) dx ( x, x ) dx dx µ µ dx dx dx dx dx

13 Expected Value and Variance Example: ( x x ) x 0,5, o. w. 0 x, 000 a) Find marginal distribution or X b) Compute E(X ) and Var(X ) Conditional Distribution Let X and Y be two random variables, discrete or continuous. The conditional distribution o the random variable Y, given that X x is ( x, y) ( y x), ( x) > 0. ( x) Similarly, the conditional distribution o the random variable X, given that Yy, is ( x, y) ( x y), ( y) > 0. ( y)

14 Conditional Distribution Example Reerring to example or the discrete case, ind conditional distribution o X, given that X, and use it to determine P(X 0 X ). Conditional Distribution Example The joint density or the random variables (X,Y), where X is the unit temperature change and Y is the proportion o spectrum shit that a certain atomic article produces is 0xy 0 < x < y < ( x, y) 0, o.w. a) Find the marginal densities (x), (y) and the conditional density (y x). b) Find the probability that spectrum shits more than hal o the total observations, given the temperature is increased to 0.5 unit.

15 Independent Random Variabes Two random variables X and Y are said to be independent i or every pair o x and y values, ( x, y) p( x) p( y) p when X and Y are discrete. ( x, y) ( x) ( y) when x and y are continuous. I this equation is not satisied or all (x,y) then x and y aresaidto be dependent. Independent Random Variabe Example Reerring to example or the discrete case, show that random variables are not statistically independent. p(0,) 30 Marginal distribution or X 4 p(0) p(0, x ) + 30 x Marginal distribution or p() 4 x 0 p( x,) 30 + X p(0,) p(0) p() X and Y are not statistically independent.

16 Independent Random Variabes Example 0xy 0 < x < y < ( x, y) 0, o.w. Show that X and Y are not statistically independent. Covariance When two random variables X and Y are not independent, it is requently o interest to measure how strongly they are related to one another. The covariance between two r.v. X and Y is; ( xy) E( x µ x)( y µ y) cov, ( µ x)( µ y) ( ) x y p x, y x, y discrete x y ( x, y) dxdy x, y continuous ( µ x)( µ y) cov, ( xy) E( xy) µ µ E( xy) E( x) E( y) x y

17 Independent Random Variabes Correlation: The correlation coeicent o X and Y denoted by Corr(X,Y), ρ xy or just only ρ is denoted by ; ρ xy Cov σ σ ( X, Y ) x y Independent Random Variabes Example The raction X o male runners and the raction Y o emale runners who compete in marathon races is described by the joint density unction 8xy ( x, y) 0, 0 x, o.w. 0 y x Find covariance o X and Y

18 Reerences Walpole, Myers, Myers, Ye, (00), Probability & Statistics or Engineers & Scientists Dengiz, B., (004), Lecture Notes on Probability, Hines, Montgomery, (990), Probability & Statistics in Engineering & Management Science