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Joint Probability Distribution The joint probability distribution function of X and Y is denoted by fXY(x,y). The marginal probability distribution function of X , fX(x) is obtained by i. summing the probabilities corresponding to each y value and the given x. (discrete case) fX(x)= SY fXY(x,y)|X=x ii. Integrating out Y from the joint pdf (continuous case) fX(x)= Y fXY(x,y)dy The conditional probability distribution function of X given Y is denoted by fX|Y(x|y) fX|Y(x|y)=fX, Y(x,y)/ fY(y) [We similarly define fY|X(y|x)] Random variables X and Y are independent if and only if fX|Y(x|y)=fX(x) for all x and y Joint Probability Distribution Covariance of two random variables X and Y Cov(X,Y) E{(X- mX)(Y-mY)} E(XY- Y mX–X mY +mXmY) E(XY) - mXE(Y) –mY E(X) + mXmY E(XY)–mXmY- mXmY + mXmY E(XY)- mXmY The Coefficient of Correlation between two random variables X and Y s(X,Y) Cov(X,Y)/ sXsY Two random variables X and Y are uncorrelated if s(X,Y) = 0; or if E(XY)=mXmY An important result: Suppose that X and Y are two random variables that have mean mX, mY and standard deviation sX and sY respectively. Then for Z aX +bY E(Z) amX + bmY VarZ a2VarX + b2VarY + 2abcov(X,Y) a2s2X+ b2 s2Y + 2absXY sX sY VarZ a2VarX + b2VarY + 2abcov(X,Y) a2s2X+ b2 s2Y + 2absXY sX sY where sXY Coefficient of Correlation between X and Y. If sXY = -1 then VarZ = (asX -bsY)2 Joint Probability Distribution Recall: Two random variables x and y are independent if their joint p.d.f. fXY (x,y) is the product of the respective marginal p.d.f. fX(x) and fY(y). That is, fXY (x,y) = fX(x). fy(y) Theorem: Independence of two random variables X and Y imply that they are uncorrelated (but the converse is not always true) Proof: E(XY) = xy fXY(x,y)dxdy E(XY) = xy g(x)h(y)dxdy E(XY) = (x g(x)dx)y h(y)dy E(XY) = (mX)y h(y)dy E(XY) = mX y h(y)dy E(XY) = mXmY The Normal Distribution A continuous distribution with the pdf: f(X) = {1/(s2p) }e –1/2[(X-mx)/sx)2 ] For the Standard Normal Distribution Variable Z, s = 1; m = 0 f(z) = {1/2p }e –(1/2) z2 Suppose that X and Y are two random variables such that they have mean mX, mY and standard deviation sX and sY respectively. Also assume that both X and Y are normally distributed. Then if W aX +bY W ~ Normal(mw, s2w) with mw amX + bmY and s2w a2s2X+ b2 s2X + 2absXY sX sY where sXY is the relevant correlation coefficient. Message: A linear combination of two or more normally distributed (independent or not) r.v.s has a normal distribution as well. The c2distribution: Consider Z ~ Normal(0,1). Consider Y = Z2. Then Y has a c2 distribution of 1 degree of freedom (d.o.f.). We write it as Y ~ c2(1) . Consider Z1, Z2, …Zn independent random variables each ~ Normal(0,1) Then their sum SZi2 has a c2 distribution with d.o.f. = n. That is, SZi2 ~ c2(n) Consider two independent random variables X ~ Normal(0,1) and Y ~ c2(n) Then the variable w X/ (Y/n) has a t-distribution with d.o.f. = n. That is, w ~ t(n) An Application Consider X ~ Normal(m, s2). Then XMEAN ~ Normal (m, s2/n) Consider X ~ (m, s2). Then XMEAN ~ Normal (m, s2/n) if n is ‘large’ (CLT) Then the variable w (XMEAN – m)/s/n), has a t-distribution with d.o.f. = n-1 where s is an unbiased estimator of s Suppose that X ~ c2(m). and Y ~ c2(n) and the variables X and Y are independent. Then the variable v ≡ (x/m) /(y/n) has an F distribution with the numerator d.o.f. = m and the denominator d.o.f = n. V ~ Fm,n Suppose that X ~ c2(1). and Y ~ c2(n) and the variables X and Y are independent. Then the variable v ≡ x /(y/n) has an F distribution with the numerator d.o.f. = 1 and the denominator d.o.f = n. V ~ F1,n Clearly, V ~ t(n)