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Joint Distributed Random Variables Joint Distributed Random Variables Joint Distribution Function (§ 6.1) Independent Random Variables (§ 6.2) Sums of Independent Random Variables (§ 6.3) Conditional Distribution Discrete Case (§ 6.4) Continuous Case (§ 6.5) Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Joint Distribution Function ♦ Joint Distribution Function Definition For any two random variables X and Y , the joint cumulative probability distribution function of X and Y is defined to be F (x, y ) = P{X ≤ x, Y ≤ y }, −∞ < x, y < ∞. Note 5.1: The joint cumulative probability distribution function satisfies the following conditions 1. F (∞, ∞) = 1. 2. F (−∞, ∞) = F (−∞, y ) = F (x, −∞) = 0. 3. If a1 ≤ a2 and b1 ≤ b2 , then F (a1 , b1 ) ≤ F (a2 , b2 ). Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Joint Distribution Function ♦ Joint Distribution Function Note 5.2: The marginal distribution functions of X and Y can be obtained from the joint cumulative probability distribution function of X and Y such that FX (x) = FY (y ) = lim F (x, y ) = F (x, ∞); n→∞ lim F (x, y ) = F (∞, y ). n→∞ Note 5.3: In theory, all joint probability statements about X and Y can be answered in terms of their joint distribution function such that P{a1 < X ≤ a2 , b1 < Y ≤ b2 } = F (a2 , b2 ) − F (a1 , b2 ) −F (a2 , b1 ) + F (a1 , b1 ). Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Joint Distribution Function ♦ Joint Distribution Function Definition The joint probability mass function for discrete random variables of X and Y is defined to be p(x, y ) = P{X = x, Y = y }. Note 5.4: The joint cumulative distribution function for discrete random variables of X and Y can be expressed by F (x, y ) = P{X ≤ x, Y ≤ y } = ∑ ∑ p(s, t). s≤x t≤y Note 5.5: The joint probability mass function of X and Y satisfies the following conditions 1. p(x, y ) ≥ 0 for every possible pair (x, y ). 2. ∑x ∑y p(x, y ) = 1 for all possible pair (x, y ). Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Joint Distribution Function ♦ Joint Distribution Function Note 5.6: The marginal probability mass functions for the univariate random variables X and Y can be derived from the joint probability mass function of X and Y such that pX (x) = P{X = x} = ∑ p(x, y ); ∑ p(x, y ). y :p(x,y )>0 pY (y ) = P{Y = y } = x:p(x,y )>0 Expected Value of g(X , Y ) Let p(x, y ) be the joint probability mass function of discrete random variables X and Y , then the expected value of a scale function g(X , Y ) is given by E g(X , Y ) = ∑ ∑ g(x, y )p(x, y ). x Qihao Xie y Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Joint Distribution Function ♦ Joint Distribution Function Example 5.1 Consider a random experiment by randomly drawn a ball from a box containing 10 balls. Each ball has an ordered pair of number on it such that (1, 1), (2, 1), (1, 2) appear on 1 ball each, (3, 1), (2, 2) appear on 2 balls each, and (3, 2) appears on 3 balls. Let X and Y be the random variables represented respectively the first and second values of the ordered pair, find (1) the joint probability mass function of X and Y , (2) pX (x) and pY (y ), and (3) E(X ) and E(Y ). Solution: Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Joint Distribution Function ♦ Joint Distribution Function Definition Given two random variables X and Y , there exists a nonnegative function f (x, y ) for all real values x and y such that for every set C of pairs of real numbers P{(X , Y ) ∈ C} = ZZ f (x, y )dxdy . (x,y )∈C Then X and Y are said to be jointly continuous random variable, and the function f (x, y ) is called the joint probability density function of X and Y . Note 5.7: Given any two sets of real number A and B, and define C = {(x, y ) : x ∈ A, y ∈ B}, then P{X ∈ A, Y ∈ B} = Z Z f (x, y )dxdy . A B Note 5.8: The joint cumulative distribution function for continuous random variables of X and Y can be expressed by F (x, y ) = P{X ≤ x, Y ≤ y } = Z x Z y f (s, t)dsdt. −∞ −∞ Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Joint Distribution Function ♦ Joint Distribution Function Note 5.9: The joint probability density function of X and Y can be derived from the joint cumulative distribution function F (x, y ) such that f (x, y ) = ∂2 F (x, y ). ∂ x∂ y Note 5.10: The joint probability density function of X and Y satisfies the following conditions 1. f (x, y ) ≥ 0 for every possible pair (x, y ). 2. R∞ R∞ −∞ −∞ f (x, y )dxdy = 1. Note 5.11: The marginal probability density functions for the univariate random variables X and Y can be derived from the joint probability density function of X and Y such that fX (x) fY (y ) Z ∞ = f (x, y )dy ; −∞ Z ∞ = Qihao Xie f (x, y )dx. −∞ Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Joint Distribution Function ♦ Joint Distribution Function Expected Value of g(X , Y ) Let f (x, y ) be the joint probability density function of continuous random variables X and Y , then the expected value of a scale function g(X , Y ) is given by Z E g(X , Y ) = ∞ Z ∞ g(x, y )f (x, y )dxdy . −∞ −∞ Example 5.2 If f (x, y ) = 6x 2 y , for 0 < x, y < 1, and 0 otherwise, find (1) P{0 < X < 34 , 31 < Y < 3}, (2) F (x, y ), (3) fX (x) and fY (y ), and (4) E(X ) and E(Y ). Solution: Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Independent Random Variables ♦ Independent Random Variables Definition The random variables X and Y are said to be independent if for any two sets of real number A and B, P{X ∈ A, Y ∈ B} = P{X ∈ A} · P{Y ∈ B}. Note 5.12: The random variables X and Y are independent if, and only if F (x, y ) = FX (x)FY (y ), where FX (x) and FY (y ) are marginal distribution of X and Y , respectively. Proof: (exercise) Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Independent Random Variables ♦ Independent Random Variables Note 5.13: The discrete random variables X and Y are independent if, and only if p(x, y ) = pX (x)pY (y ), where pX (x) and pY (y ) are marginal distribution of X and Y , respectively. Proof: Example 5.3 The random variables of X and Y in Example 5.1 are not independent. Solution: Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Independent Random Variables ♦ Independent Random Variables Note 5.14: The continuous random variables X and Y are independent if, and only if f (x, y ) = fX (x)fY (y ), where fX (x) and fY (y ) are marginal distribution of X and Y , respectively. Proof: (exercise) Example 5.4 The random variables of X and Y in Example 5.2 are independent. Solution: Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Independent Random Variables ♦ Independent Random Variables Proposition 5.1 Two random variables X and Y are independent if, and only if there exist two functions g(x) and h(y ) such that 1. The joint probability mass function can be expressed as p(x, y ) = g(x)h(y ) for eveny x ∈ R, y ∈ R. 2. The joint probability density function can be expressed as f (x, y ) = g(x)h(y ) for eveny x ∈ R, y ∈ R. Proof: Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Sums of Independent Random Variables ♦ Sums of Independent Random Variables Proposition 5.2 Suppose that X and Y are two independent continuous random variables having probability density functions fX (x) and fY (y ), respectively. Then, the probability density function of the random variable Z = X + Y is given by fZ (z) = Z ∞ −∞ fX (z − y )fY (y )dy . Notice that: fZ is called the convolution of the density functions fX (x) and fY (y ). Corollary 5.1 If X ∼ Poi(λ1 ) and Y ∼ Poi(λ2 ) are two independent random variables, then Z ∼ Poi(λ1 + λ2 ). Proof: Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Sums of Independent Random Variables ♦ Sums of Independent Random Variables Corollary 5.2 If X ∼ Exp(λ ) and Y ∼ Exp(λ ) are two independent random variables, then Z ∼ Gamma(2, λ ). Proof: Corollary 5.3 If X ∼ Gamma(α1 , λ ) and Y ∼ Gamma(α2 , λ ) are two independent random variables, then Z ∼ Gamma(α1 + α2 , λ ). Proof: Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Sums of Independent Random Variables ♦ Sums of Independent Random Variables Corollary 5.4 If X ∼ N(µ1 , σ12 ) and Y ∼ N(µ2 , σ22 ) are two independent random variables, then Z ∼ N(µ1 + µ2 , σ12 + σ22 ). Proof: Corollary 5.5 Given an independent random variable X1 , . . . , Xn , each follows a normal distribution with mean µi and variance σi2 , for i = 1, . . . , n, then n n n ∑ Xi ∼ N ∑ µi , ∑ σi2 . i=1 i=1 i=1 Proof: (exercises) Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Conditional Distribution ♦ Conditional Distribution DISCRETE CASE Definition If X and Y is a joint discrete random variables having a joint probability mass function p(x, y ), then the conditional probability mass function of X given Y = y is defined to be p(x, y ) pX |Y (x|y ) = , pY (y ) provided pY (y ) > 0 is the marginal probability mass function of Y . Note 5.15: Suppose that pX |Y (x|y ) is the conditional probability mass function of X given Y = y , then the conditional cumulative distribution of X given Y = y is FX |Y (x|y ) = P{X ≤ x|Y = y } = ∑ pX |Y (t|y ). t≤x Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Conditional Distribution ♦ Conditional Distribution Note 5.16: If X and Y are independent discrete random variables, then pX |Y (x|y ) = pX (x) and pY |X (y |x) = pY (y ). Proof: Example 5.5 Find the pX |Y (x|y ) for the random variables given in Example 5.1. Solution: Qihao Xie Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Conditional Distribution ♦ Conditional Distribution CONTINUOUS CASE Definition If X and Y is a joint continuous random variables having a joint probability density function f (x, y ), then the conditional probability density function of X given Y = y is defined to be f (x, y ) , fX |Y (x|y ) = fY (y ) provided fY (y ) > 0 is the marginal probability density function of Y . Note 5.19: Suppose that fX |Y (x|y ) is the conditional probability density function of X given Y = y , then the conditional cumulative distribution of X given Y = y is FX |Y (x|y ) = P{X ≤ x|Y = y } = Qihao Xie Z x −∞ fX |Y (t|y )dt. Introduction to Probability and Basic Statistical Inference Joint Distributed Random Variables ⇒ Conditional Distribution ♦ Conditional Distribution Note 5.20: If X and Y are independent continuous random variables, then fX |Y (x|y ) = fX (x) and fY |X (y |x) = fY (y ). Proof: Example 5.6 Find the fX |Y (x|y ) for the random variables given in Example 5.2. Solution: Qihao Xie Introduction to Probability and Basic Statistical Inference