Joint Densities and distributions derived fromthem (I)Juana [email protected] Department of StatisticsJ. Sanchez Joint densities1. Intro ductionWe can have joint distribution functions associated with two or morerandom variables. We will study them for two random variables.Joint probability density functionThe joint density of a pair of continuous random variables X and Y is f (X , Y ),and it has the following properties:(a) The joint density is always nonnegative, i.e.,f (x , y) ≥ 0 for all X , Y(b) If A and B are sets of real numbers, thenP(X ∈ A andY ∈ B ) =ZAZBf (x , y)dydx, that is the volume under f (X , Y ) in the region defined by A and B.(c) The double integral of f (X, Y ) over the whole domain of f (X, Y ) is 1.J. Sanchez Joint densitiesExampleLet X be the time that Alice waitsfor a traffic light to turn green,and let Y be the time (at adifferent intersection) that Bobwaits for a traffic light to turngreen. Suppose that X and Yhave joint densityf (x, y) = 15e−3x−5yx ≥ 0, y ≥ 0(a) Is f (x, y ) a density?(b) Calculate theP(X ≤ 2, Y ≤ 6)(c) Calculate the probability thatBob waits longer at his lightthan Alice.Figure :f (x , y) = 15e−3x−5yx > 0, y > 0J. Sanchez Joint densitiesExample (cont’d)f (x, y) = 15e−3x−5yx > 0, y > 0Is f (x, y) a density? YES, weshow.R∞0R∞015e−3x−5ydydx =R∞03e−3xR∞05e−5ydy dx = 1The integral inside the brackets isthe integral of an exponentialdensity with parameter λ = 5 andthe integral outside is the integralof an exponential density withparameter λ = 3 Each of those is1.Figure :f (x , y) = 15e−3x−5yx > 0, y > 0J. Sanchez Joint densitiesExample 15(cont’d)f (x, y) = 15e−3x−5yx > 0, y > 0Calculate the P(X ≤ 2, Y ≤ 6)R20R6015e−3x−5ydydx =R203e−3xhR605e−5ydyidx =1 − e−61 − e−30The integral inside the brackets isthe cumulative F(y=6) of anexponential with λ = 5 and theintegral left after finding that isthe cumulative F(x=2) of anexponential with parameter λ = 3Figure :f (x , y) = 15e−3x−5yx > 0, y > 0J. Sanchez Joint densities2 Probabilities of relations between two random variablesThe third question asked in example 15.1.1 is the probability of a relationbetween the two random variables, P(Y > X ). We will integrate firstover all the y values.f (x, y) = 15e−3x−5yx > 0, y > 0Calculate the probability that Bobwaits longer at his light than Alice.P(Y > X ) =R∞0Ry015e−3x−5ydxdy =R∞0−5e−3x−5y|yx=0dy =R∞0(5e−5y− 5e−8y)dy = 3/8Figure : Setting up integral for P(Y > X), with y as theouter integral and x as the inner integral. Fixed value of y (here,for example, y=3.2), and x ranging from 0 to y .J. Sanchez Joint densitiesWe can do the same problem integrating over all values of x. P(Y > X ).f (x, y) = 15e−3x−5yx > 0, y > 0Calculate the probability that Bobwaits longer at his light than Alice.P(Y > X ) =R∞0R∞x15e−3x−5ydydx =R∞0−3e−3x−5y|∞y=xdx =R∞0(3e−3x−5x)dx = 3/8Figure : Setting up integral for P(Y > X), with x as theouter integral and y as the inner integral. Fixed value of x (here,for example, x=2.6), and y ranging from x to ∞.J. Sanchez Joint densities3 Independent Continuous Random VariablesIndependent continuous random variablesTwo continuous random variables X and Y are independent if their jointdensity is equal to the product of their univariate densities, i.e., iff (x , y) = f (x )f (y ),for all x and yJ. Sanchez Joint densities4. Marginal densities. Calculating the density of a randomvariable when only the joint density is givenIn order to determine independence of two random variables X and Y weneed to be able to find the density of X and the density of Y .Calculating the density of a random variable when only the joint density is givenIf two continuous random variables have joint density f (x , y ), then the densityof X can be calculated as follows:f (x ) =Z∞−∞f (x , y)dy. In order to get the density of X , we integrate over all of the possible y values,which can be thought of (informally) as integrating y out of the picture. Theresulting density is called the marginal density of X .To find f (y ) do the same, but integrate with respect to x.J. Sanchez Joint densitiesExampleA certain process for production of an industrial chemical yields aproduct that contains two main types of impurities. For a certain volumeof sample from this process, let X denote the proportion of totalimpurities in the sample, and let Y denote the proportion of type Iimpurity among all impurities found. Suppose that under investigation ofmany such samples, the join distribution of X and Y can be adequatelymodeled by the following functionf (x, y) = 2(1 − x ) 0 ≤ x ≤ 1 0 ≤ y ≤ 1f (x) =R102(1 − x )dy = 2(1 − x) 0 < x < 1f (y) =R102(1 − x )dx = 1 0 < y < 1Check independence: Is f(x,y) = f(x)f(y) ? YES. So x, y are independent.J. Sanchez Joint densitiesExampleGasoline is to be stocked in a bulk tank once each week and then sold tocustomers. Let X denote the proportion of the tank that is stocked in aparticular week, and let Y denote the proportion of the tank that is soldin the same week. Due to limited supplies, X is not fixed in advance butvaries from week to week. Suppose that a study of many weeks showsthe joint relative frequency behavior of X and Y to be such that the joindensity function provides and adequate model:f (x, y) = 3x 0 < y < x < 1f (y) =R∞−∞f (x, y)dx=R1y3xdx =32x21y=32(1 − y2) 0 ≤ y ≤ 1f (x) =Zx03xdy = 3x2; 0 < x < 1Because 3x2(32(1 − y )2) 6= 3x the random variables X and Y are notindependent.J. Sanchez Joint densities5. Determining the joint density f (x, y) from f (x) andf (y )Joint Density of Independent Random VariablesIf X is a random variable with pdf f (x ) and Y is a random variable with pdff (y ), and we know that X and Y are independent, then we can write the jointdensity f (x, y ) as f (x, y) = f (x )f (y ).If we have many independent random variables, say X1, X2, ...., Xnand we knowf (xi) for all i, then the multivariate joint densityf (x1, x2, ....., xn) = f (x1)f (x2)......f (xn)This result is very important in the application of probability to StatisticsJ. Sanchez Joint densitiesExampleThe lifetime of a music player (before it permanently fails) is a randomvariable X with densityf (x) =13e−x/3x > 0Consider two music players that are assumed to have independentlifetimes, X and Y , respectively, but the same pdf family . Then thejoint
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