06/05/2008 Jae Hyun Kim Chapter 2 Probability Theory (ii) : Many Random Variables Bioinformatics Tea Seminar: Statistical Methods in Bioinformatics

Joint Probability Distribution Joint Density Function Independence Generating Functions Covariance and Correlation Marginal Distribution Conditional Distribution 2 Content

Joint Probability Distribution Joint Density Function 3 Joint Probability

Y 1, Y 2,..., Y n are discrete random variables and said to be independent if Continuous random variables X 1, …, X n are independent if 4 Independence

Sum of Random Variables Pgf If pgf of Y i = p(t) Mgf If X 1, …, X n are iid, 5 Generating Functions

Mean and Variance (iid) If X1, …, Xn are independent, 6 Generating Functions

Covariance Correlation  a measure of the degree of linear association between the random variables involved. 7 Dependent Case

X, Y are independent   Correlation = 0 Possibly Correlation = 0 even if X, Y are dependent Correlation = 1 if Y = aX + b (a>0) Correlation = -1 if Y = aX + b (a<0) 8 Correlation

Marginal Probability Distribution of Y 1 ignoring Y 2 Discrete case Continuous case 9 Marginal Distribution

Conditional Probability of Y 2 = y 2, given that Y 1 = y 1 Discrete case Continuous case 10 Conditional Probability

Definition If X 1 and X 2 are independent random variables, discrete or continuous, then 11 Expected Value

Mean Variance Independent case 12 Mean and Variance

Convergence in distribution  Convergence of sequence of cumulative density function F i (x) for X 1, …, X n, …  Sequence {X n } converges to X if 13 Asymptotic Distribution