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Theorem: 7.1. p 191 A real valued function f of two variables is joint probability density function of a pair of discrete random variables X and Y if and only if : (1)f XY (x , y ) 0 for all ( x , y ) (2) f XY (x , y ) 1 x y Example:7.1 page 191 For what value of the constant k the function given by if x 1, 2,3; y 1, 2,3 k x y f (x , y ) otherwise 0 Is a joint probability density function of some random variables X , Y ? Marginal probability density function Example: : المحاضرة الثالثة10/4/1435 اإلثنين 7.2. Bivariate Continuous Random Variables 7.2. Bivariate Continuous Random Variables In this section, we shall extend the idea of probability density functions of one random variable to that of two random variables. Definition 7.5. The joint probability density function of the random variables X and Y is an integrable function f(x, y) such that (1)f X Y (x , y ) 0 for all ( x , y ) (2) f X Y ( x , y ) dx dy 1 2 7.2. Bivariate Continuous Random Variables Example 7.6. Let the joint density function of X and Y be given by k x y 2 f (x , y ) 0 if 0 x y 1 otherwise What is the value of the constant k ? REMARK: If we know the joint probability density function f of the random variables X and Y , then we can compute the probability of the event A from: P ( A ) A f X Y (x , y ) dx dy Bivariate Continuous Random Variables Example 7.7. Let the joint density of the ontinuous random variables X and Y be 6 2 if 0 x 1;0 y 1 (x x y f (x , y ) 5 0 otherwise What is the probability of the event ( x y )? Marginal probability density function: Definition 7.6. Let (X, Y ) be a continuous bivariate random variable. Let f(x, y) be the joint probability density function of X and Y . The function f 1 (x ) f (x , y ) dy is called the marginal probability density function of X. Similarly, the function Marginal probability density function: Similarly, the function f 1( y ) f (x , y ) dx is called the marginal probability density function of Y. Marginal probability density function: Example 7.9. If the joint density function for X and Y is given by: 2e x y if 0 x y f (x , y ) otherwise 0 What is the marginal density of X where nonzero? Definition 7.7. Let X and Y be the continuous random variables with joint probability density function f(x, y). The joint cumulative distribution function F(x, y) of X and Y is defined as y F ( x , y ) P ( X x ,Y y ) x for all ( x , y ) 2 . f XY (u ,v ) du dv The joint cumulative distribution function F(x, y): From the fundamental theorem of calculus, we again: obtain 2F ( x , y ) f (x , y ) . x y The joint cumulative distribution function F(x, y): Example 7.11. If the joint cumulative distribution function of X and Y is given by 1 3 2 2 (2 x y 3 x y ) for 0 x , y 1 F (x , y ) 2 0 otherwise then what is the joint density of X and Y ? EXERCISES: Page 208-209 1 , 2 , 3 , 4 , 7 , 8 , 10 , 11 7.3. Conditional Distributions First, we motivate the definition of conditional distribution using discrete random variables and then based on this motivation we give a general definition of the conditional distribution. Let X and Y be two discrete random variables with joint probability density f(x, y). 7.3. Conditional Distributions Then by definition of the joint probability density, we have f(x, y) = P(X = x, Y = y). If A = {X = x}, B = {Y = y} and f (y) = P(Y = y), then from the above equation we have P ({X = x} / {Y = y}) = P (A/B) P (A B P (B ) P { X x }and P {Y y } f ( x , y ) P {Y y } f (y ) 7.3. Conditional Distributions If we write the P ({X = x} / {Y = y}) as g(x / y), then we have f (x , y ) g (x / y ) f (y ) 7.3. Conditional Distributions Definition 7.8. Let X and Y be any two random variables with joint density f(x, y) and marginals f1(x) and f2(y). The conditional probability density function g of X, given (the event) Y = y, is defined as f (x , y ) g (x / y ) , f ( y ) 0 f (y ) 7.3. Conditional Distributions Similarly, the conditional probability density function h of Y , given (the event) X = x, is defined as f (x , y ) h (x / y ) , f (x ) 0 f (y ) 7.3. Conditional Distributions Example 7.14. Let X and Y be discrete random variables with joint probability function 1 ( x y ) for x 1, 2,3; y 1, 2 f ( x , y ) 21 otherwise 0 What is the conditional probability density function of Y, given X = 2 ? 7.3. Conditional Distributions Example 7.15. Let X and Y be discrete random variables with joint probability function ( x y ) f ( x , y ) 32 0 for x 1, 2; y 1, 2, 3, 4 otherwise What is the conditional probability density function of Y, given X = x ? 7.3. Conditional Distributions Example 7.16. Let X and Y be contiuous random variables with joint pdf 12 x f (x , y ) 0 for 0 y 2x 1 otherwise What is the conditional probability density function of Y, given X = x ? 7.3. Conditional Distributions Example 7.17. Let X and Y random variables such that X has pdf 24 x f (x ) 0 1 2 otherwise for 0 x 2 and the conditional density of Y given X = x is y h (x / y )2x 0 2 , for 0 y 2x otherwise 7.3. Conditional Distributions What is the conditional density of X given Y = y over the appropriate domain? 7.4. Independence of Random Variables In this section, we define the concept of stochastic independence of two random variables X and Y . The conditional robability density function g of X given Y = y usually depends on y. If g is independent of y, then the random variables X and Y are said to be independent. This motivates the following definition. 7.4. Independence of Random Variables Definition 7.8. Let X and Y be any two random variables with joint density f(x, y) and marginals f1(x) and f2(y). The random variables X and Y are (stochastically) independent if and only if f ( x , y ) f ( x ) f ( y ) for all ( x , y ) x y . 7.4. Independence of Random Variables Example 7.18. Let X and Y be discrete random variables with joint density 1 36 f (x , y ) 2 36 for 1 x y 6 for 1 x y 6 Are X and Y stochastically independent? 7.4. Independence of Random Variables Example 7.19. Let X and Y have the joint density e ( x y ) for 0 x , y f (x , y ) otherwise 0 Are X and Y stochastically independent? 7.4. Independence of Random Variables Example 7.20. Let X and Y have the joint density x y f (x , y ) 0 for 0 x 1 ; 0 y 1 otherwise Are X and Y stochastically independent? 7.4. Independence of Random Variables Definition 7.9. The random variables X and Y are said to be independent and identically distributed (IID) if and only if they are independent and have the same distribution. EXERCISES: Page 210-211 14 , 16 , 21