Independence of random variables Definition Random variables X and Y are independent if their joint distribution function factors into the product of their marginal distribution functions Theorem Suppose X and Y are jointly continuous random variables. X and Y are independent if and only if given any two densities for X and Y their product is the joint density for the pair (X,Y) i.e. Proof: If X and Y are independent random variables and Z =g(X), W = h(Y) then Z, W are also independent. week 8

Example Suppose X and Y are discrete random variables whose values are the non-negative integers and their joint probability function is Are X and Y independent? What are their marginal distributions? Factorization is enough for independence, but we need to be careful of constant terms for factors to be marginal probability functions. week 8

Example and Important Comment The joint density for X, Y is given by Are X, Y independent? Independence requires that the set of points where the joint density is positive must be the Cartesian product of the set of points where the marginal densities are positive i.e. the set of points where fX,Y(x,y) >0 must be (possibly infinite) rectangles. week 8

Conditional densities If X, Y jointly distributed continuous random variables, the conditional density function of Y | X is defined to be if fX(x) > 0 and 0 otherwise. If X, Y are independent then . Also, Integrating both sides over x we get This is a useful application of the law of total probability for the continuous case. week 8

Example Consider the joint density Find the conditional density of X given Y and the conditional density of Y given X. week 8

Properties of Expectations Involving Joint Distributions For random variables X, Y and constants E(aX + bY) = aE(X) + bE(Y) Proof: For independent random variables X, Y E(XY) = E(X)E(Y) whenever these expectations exist. week 8

Covariance Recall: Var(X+Y) = Var(X) + Var(Y) +2 E[(X-E(X))(Y-E(Y))] Definition For random variables X, Y with E(X), E(Y) < ∞, the covariance of X and Y is Covariance measures whether or not X-E(X) and Y-E(Y) have the same sign. Claim: Proof: Note: If X, Y independent then E(XY) =E(X)E(Y), and Cov(X,Y) = 0. week 8

Example Suppose X, Y are discrete random variables with probability function given by Find Cov(X,Y). Are X,Y independent? y x -1 1 pX(x) 1/8 pY(y) week 8

Important Facts Independence of X, Y implies Cov(X,Y) = 0 but NOT vice versa. If X, Y independent then Var(X+Y) = Var(X) + Var(Y). If X, Y are NOT independent then Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y). Cov(X,X) = Var(X). week 8

Example Suppose Y ~ Binomial(n, p). Find Var(Y). week 8

Properties of Covariance For random variables X, Y, Z and constants Cov(aX+b, cY+d) = acCov(X,Y) Cov(X+Y, Z) = Cov(X,Z) + Cov(Y,Z) Cov(X,Y) = Cov(Y, X) week 8

Correlation Definition For X, Y random variables the correlation of X and Y is whenever V(X), V(Y) ≠ 0 and all these quantities exists. Claim: ρ(aX+b,cY+d) = ρ(X,Y) Proof: This claim means that the correlation is scale invariant. week 8

Theorem For X, Y random variables, whenever the correlation ρ(X,Y) exists it must satisfy -1 ≤ ρ(X,Y) ≤ 1 Proof: week 8

Interpretation of Correlation ρ ρ(X,Y) is a measure of the strength and direction of the linear relationship between X, Y. If X, Y have non-zero variance, then . If X, Y independent, then ρ(X,Y) = 0. Note, it is not the only time when ρ(X,Y) = 0 !!! Y is a linearly increasing function of X if and only if ρ(X,Y) = 1. Y is a linearly decreasing function of X if and only if ρ(X,Y) = -1. week 8

Example Find Var(X - Y) and ρ(X,Y) if X, Y have the following joint density week 8

Conditional Expectation For X, Y discrete random variables, the conditional expectation of Y given X = x is and the conditional variance of Y given X = x is where these are defined only if the sums converges absolutely. In general, week 8

For X, Y continuous random variables, the conditional expectation of Y given X = x is and the conditional variance of Y given X = x is In general, week 8

Example Suppose X, Y are continuous random variables with joint density function Find E(X | Y = 2). week 8

More about Conditional Expectation Assume that E(Y | X = x) exists for every x in the range of X. Then, E(Y | X ) is a random variable. The expectation of this random variable is E [E(Y | X )] Theorem E [E(Y | X )] = E(Y) This is called the “Law of Total Expectation”. Proof: week 8

Example Suppose we roll a fair die; whatever number comes up we toss a coin that many times. What is the expected number of heads? week 8

Theorem For random variables X, Y V(Y) = V [E(Y|X)] + E[V(Y|X)] Proof: week 8

Example Let X ~ Geometric(p). Given X = x, let Y have conditionally the Binomial(x, p) distribution. Scenario: doing Bernoulli trails with success probability p until 1st success so X : number of trails. Then do x more trails and count the number of success which is Y. Find, E(Y), V(Y). week 8

Law of Large Numbers Toss a coin n times. Suppose Xi’s are Bernoulli random variables with p = ½ and E(Xi) = ½. The proportion of heads is . Intuitively approaches ½ as n  ∞ . week 8

Markov’s Inequality If X is a non-negative random variable with E(X) < ∞ and a >0 then, Proof: week 8

Chebyshev’s Inequality For a random variable X with E(X) < ∞ and V(X) < ∞, for any a >0 Proof: week 8

Back to the Law of Large Numbers Interested in sequence of random variables X1, X2, X3,… such that the random variables are independent and identically distributed (i.i.d). Let Suppose E(Xi) = μ , V(Xi) = σ2, then and Intuitively, as n  ∞, so week 8

Formally, the Weak Law of Large Numbers (WLLN) states the following: Suppose X1, X2, X3,…are i.i.d with E(Xi) = μ < ∞ , V(Xi) = σ2 < ∞, then for any positive number a as n  ∞ . This is called Convergence in Probability. Proof: week 8

Example Flip a coin 10,000 times. Let E(Xi) = ½ and V(Xi) = ¼ . Take a = 0.01, then by Chebyshev’s Inequality Chebyshev Inequality gives a very weak upper bound. Chebyshev Inequality works regardless of the distribution of the Xi’s. week 8

Strong Law of Large Number Suppose X1, X2, X3,…are i.i.d with E(Xi) = μ < ∞ , then converges to μ as n  ∞ with probability 1. That is This is called convergence almost surely. week 8