Understanding Random Variables and Distributions in Statistics

Unit 2 : Random Variables and their Distributions Wenyaw Chan Division of Biostatistics School of Public Health University of Texas - Health Science Center at Houston

Random Variable Random Variable: A numeric function that assigns probabilities to different events in a sample. Discrete Random Variable: A random variable that assumes only a finite or denumerable number of values. The probability mass function of a discrete random variable X that assumes values x1, x2, is p(x1), p(x2), ., where p(xi)=Pr[X= xi]. Continuous Random Variable: A random variable whose possible values cannot be enumerated.

Example: Flip a coin 3 times Random Variable X = # of heads in the 3 coin tosses Probability Mass Function P(X=3) = P{(HHH)} =1/8 P(X=2) = P{HHT, HTH, THH}= 3/8 P(X=1) = P{HTT,THT, TTH} = 3/8 P(X=0) = P{TTT} = 1/8 X is a discrete random variable with probability (mass) function x 0 1 2 3 P(X=x) 1/8 3/8 3/8 1/8

Random Variable Expected value of X : k = = = = i ( ) Pr( ) E X x X x i i 1 Variance of X : Var k 2 2 = = = ( ) X = i ( ) Pr( ) X x x i i 1 Standard Deviation of X: = Var (X )

Random Variable Note : Var 2 = = 2 ( ) ( ) X E X 2 ( ) [ ( )] E X E X Cumulative Distribution Function of X : Pr(X<=x) = F(x)

Binomial Distribution Examples of the binomial distribution have a common structure: n independent trials each trial has only two possible outcomes, called success and failure . Pr (success) = p for all trials

Binomial Distribution If X= # of successful trials in these n trials, then X has a binomial distribution. ( ) k n k n k = = 1 ( ) P X k p p k=0,1,2, .,n = n k ! n where n k k ( )! ! Example: Flip a coin 10 times

Properties of Binomial Distribution If X~ Binomial (n, p), then E(X) = np Var (X) = np(1-p)

Poisson Distribution k e ( ) = = Pr X k ! k k=0,1,2, .. If X~ Poisson ( ), then EX = and VarX =

Poisson Process Assumption 1: Pr {1 event occurs in a very small time interval [0, t)} t Pr {0 event occurs in a very small time interval [0, t)} 1- t Pr{more than one event occurs in a very small time interval [0, t)} 0 Assumption 2: Probability that the number of events occur per unit time is the same through out the entire time interval Assumption 3: Pr {one event in [t1,t2) | one event in [t0, t1)} = Pr {one event in [t1, t2)}

Poisson Distribution X=The number of events occurred in the time period t for the above process with parameter , then mean= t and ( ) ( Pr k t k ) e t = = = X k ! where k= 0,1,2, and e= 2.71828 E(X)=Var(X)= t

Poisson approximation to Binomial If X~ Binomial (n, p), n is large and p is small, then e k X P = np k ( ) np ( ) ! k

Continuous Probability Distributions Probability density function (p.d.f.) (of a random variable): a curve such that the area under the curve between any two points a and b, equals Prob[a x b ]= a x bf(x)dx Pr(a<=X<=b) a b

Continuous Probability Distributions Cumulative distribution function: Pr(x a) Pr(X<=a) a

Continuous Probability Distributions The expected value of a continuous random variable X is xf(x)dx, where f(x) is the p.d.f. of X. The definition for the variance of a continuous random variable is the same as that of a discrete random variable, i.e. Var(X)=E(X2)- (EX)2= (x- )2f(x)dx, where =E(X).

The Normal Distribution (The Gaussian distribution) The p.d.f. of a normal distribution 1 ) ( = = x f , - < x < 1 exp 2 ( ) x 2 2 2

The Normal Distribution point of inflection s s u-s u u+s figures: a bell-shaped curve symmetric about Notation: X~N( , 2) : mean 2: variance

The Normal Distribution N(0,1) is the standard normal distribution If X~ N(0,1), then Pr( ) ( x = ) X x ~ : is distributed as , : c.d.f. for the standard normal r.v. Note: The point of inflection is a point where the slope of the curve changes its direction.

Properties of the N(0,1) 1. (-x) = 1- (x) 2. About 68% of the area under the standard normal curve lies between 1 and 1. About 95% of the area under the standard normal curve lies between 2 and 2. About 99% of the area under the standard normal curve lies between 2.5 and 2.5.

Properties of the N(0,1) If X~ N(0,1) and P(X< Zu)=u, 0 u 1 then Zuis called the 100 uthpercentile of the standard normal distribution. 95th%tile=1.645, 97.5th%tile=1.96, 99th%tile=2.33 Area=u Zu

Properties of the N(0,1) If X~ N( , 2), then X ~ (0,1) N This property allows us to calculate the probability of a non-standard normal random variable. a X b = P a X b P r r b a =

Other Distributions--- t distribution Let X1, .Xnbe a random sample from a normal population N( , 2). Then X s n / has a t distribution with n-1 degrees of freedom (df).

Other Distributions--- Chi-square distribution Let X1, .Xnbe a random sample from a normal population N(0, 1). Then n = 2 i X 1 i has a chi-square distribution with n degrees of freedom (df).

Other Distributions--- F distribution Let U and V be independent random variables and each has a chi-square distribution with p and q degrees of freedom respectively. Then / / V q U p has a F distribution with p and q degrees of freedom (df).

Covariance and Correlation The covariance between two random variables is defined by Cov(X,Y)=E[(X- X)(Y- Y)]. The correlation coefficient between two random variables is defined by =Corr(X,Y)=Cov(X,Y)/( X Y).

Variance of a Linear Combination Var(c1X1 + c2X2) 2 1 c c + 2 1 c c + 2 2 X 2 2 = + ( ) ( ) c Var X c Var X 1 2 2 ( , ) Cov X + 1 Var 2 1 2 = ( ) ( ) c X c Var X 1 2 X X 2 ( , ) Corr X 1 2 1 2 Y