... DISCRETE random variables X, Y Joint Probability Mass Function y1

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Presentation transcript:

... DISCRETE random variables X, Y Joint Probability Mass Function y1 … yc X x1 p(x1, y1) p(x1, y2) ... p(x1, yc) x2 p(x2, y1) p(x2, y2) p(x2, yc) xr p(xr, y1) p(xr, y2) p(xr, yc)

... DISCRETE random variables X, Y Joint Probability Mass Function y1 … yc X x1 p(x1, y1) p(x1, y2) ... p(x1, yc) pX (x1) x2 p(x2, y1) p(x2, y2) p(x2, yc) pX (x2) xr p(xr, y1) p(xr, y2) p(xr, yc) pX (xr) pY (y1) pY (y2) pY (yc) 1

Y = X =

X and Y are not independent! Probabilities… X and Y are not independent!

X and Y are not independent! Probabilities… X and Y are not independent! cdf

X and Y are not independent! Probabilities… X and Y are not independent! cdf

Cumulative Probability X = Event T = t Outcomes (AM, PM) Probability Cumulative Probability 2 (1, 1) .25 3 (1, 2), (2, 1) .45 = .20 + .25 .70 = .25 + .45 4 (1, 3), (2, 2) .25 = .15 + .10 .95 = .70 + .25 5 (2, 3) .05 1.00 = .95 + .05

... DISCRETE random variables X, Y Joint Probability Mass Function y1 … yc X x1 p(x1, y1) p(x1, y2) ... p(x1, yc) pX (x1) x2 p(x2, y1) p(x2, y2) p(x2, yc) pX (x2) xr p(xr, y1) p(xr, y2) p(xr, yc) pX (xr) pY (y1) pY (y2) pY (yc) 1

... CONTINUOUS random variables X, Y Joint Probability Density Function Joint Probability Mass Function Y y1 y2 … yc X x1 p(x1, y1) p(x1, y2) ... p(x1, yc) pX (x1) x2 p(x2, y1) p(x2, y2) p(x2, yc) pX (x2) xr p(xr, y1) p(xr, y2) p(xr, yc) pX (xr) pY (y1) pY (y2) pY (yc) 1

Joint Probability Density Function CONTINUOUS Joint Probability Density Function Volume under density f(x, y) over A. “area element” Area A

... Corollary ~ CONTINUOUS random variables X, Y Joint Probability Density Function Joint Probability Mass Function Y y1 y2 … yc X x1 p(x1, y1) p(x1, y2) ... p(x1, yc) pX (x1) x2 p(x2, y1) p(x2, y2) p(x2, yc) pX (x2) xr p(xr, y1) p(xr, y2) p(xr, yc) pX (xr) pY (y1) pY (y2) pY (yc) 1 Corollary ~ If X and Y are independent, then the joint cdf satisfies Proof: Exercise

Extension to multiple random variables X1, X2, X3,…, Xn For simplicity, take n = 3: Discrete Continuous (e.g, Multinomial distribution)