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MULTIPLE CONTINUOUS r.v. JCFD (joint CDF) for r.v X & Y:F X,Y (x,y) = P[X < x, Y < y] JPDF :. Marginal pdf. X, Y r.v’s with JPDF f X,Y (x,y). the marginal pdf of X is just f X (x) : Similarly for the marginal pdf of Y:. Functions of 2 r.v’s. - PowerPoint PPT Presentation
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Chpt. 5 1
MULTIPLE CONTINUOUS r.v
• JCFD (joint CDF) for r.v X & Y:FX,Y(x,y) = P[X < x, Y < y]
• JPDF : fX ,Y x,y 2FX ,Y x, y
x y
Chpt. 5 2
Marginal pdf
• X, Y r.v’s with JPDF fX,Y(x,y).
• the marginal pdf of X is just fX(x) :
• Similarly for the marginal pdf of Y:
fX x fX,Y x,y dy
fY y fX ,Y x, y dx
Chpt. 5 3
Functions of 2 r.v’s
• 2 r.v’s X, Y with JPDF fX,Y(x,y). We
may be interested in some functionW=g(X,Y)
• To determine the corresponding pdf and cdf of W : fW(w), and FW(w).
Chpt. 5 4
Expectation Values for 2 r.v’s X & Ywith respect to JPDF fX,Y(x,y)
• E[W]=E[g(X,Y)] - w.r.t JPDF fX,Y(x,y)
• E[X+Y] = E[X] +E[Y]
• COVARIANCE
• CORRELATION COEFFICIENT
Cov X,Y E X.Y X Y
1 X ,Y Cov X,Y X .Y
1
Chpt. 5 5
CONDITIONAL PDF
•
• conditional expectation value of g(X,Y) given y : will use conditional pdf fX|Y(x|y)
fY |X y | x fX ,Y x, y fX x
Chpt. 5 6
INDEPENDENT R.V’s
• X, Y are independent
• X, Y indep. then Cov[X,Y] = 0
– E[ g(X). h(Y) ] = E[g(X)] . E[h(Y)]
– Var[X+Y] = Var[X] + Var[Y]
fX,Y x,y fX x . fY y
Chpt. 5 7
JOINTLY GAUSSIAN R.V’s
• Jointly Gaussian : if all the marginal PDF’s of the r.v’s are Gaussian
• bivariate Gaussian PDF (2 r.v’s) :– marginal PDF’s fX(x), fY(y)
– conditional PDF of Y given X
– correlation coefficient X,Y