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The Binomial random variable A random variable that has the following pmf is said to be a binomial random variable with parameters n, p n i p(i ) p (1 p) n i i n n! where n Ai , (n i )!i ! i n is an integer 1 and 0 p 1. Example: A series of n independent trials, each having a probability p of being a success and 1 – p of being a failure, are performed. Let X be the number of successes in the n trials. The Poisson random variable A random variable that has the following pmf is said to be a Poisson random variable with parameter l (l > 0) p(i) P{ X i} e l li i! , i 0,1,.... Example: The number of cars sold per day by a dealer is Poisson with parameter l = 2. What is the probability of selling no cars today? What is the probability of selling 2? Example: The number of cars sold per day by a dealer is Poisson with parameter l = 2. What is the probability of selling no cars today? What is the probability of receiving 100? Solution: P(X=0) = e-2 0.135 P(X = 2)= e-2(22 /2!) 0.270 Continuous random variables We say that X is a continuous random variable if there exists a non-negative function f(x), for all real values of x, such that for any set B of real numbers, we have P{ X B} f ( x)dx. B The function f(x) is called the probability density function (pdf) of the random variable X. Properties P{ X (, )} f ( x)dx 1. P{a x a } a a f ( x )dx f (a ). b P{a x b} f ( x)dx a a P{ X a} f ( x)dx 0 a F (a) P( x a ) a dF (a ) f (a) da f ( x )dx The Uniform random variable A random variable that has the following pdf is said to be a uniform random variable over the interval (a, b) 1 f ( x) b a 0 if a x b otherwise. The Uniform random variable A random variable that has the following pdf is said to be a uniform random variable over the interval (a, b) 1 f ( x) b a 0 0 xa F ( x) ba 1 if a x b otherwise. if x a if a x b if x b. The Exponential random variable A random variable that has the following pdf is said to be a exponential random variable with parameter l > 0 l e - l x if x 0 f ( x) if x 0 0 The Exponential random variable A random variable that has the following pdf is said to be a exponential random variable with parameter l > 0 l e - l x f ( x) 0 x if x 0 if x 0 F ( x) l e- lt dt 1 e l x , 0 x 0. The Gamma random variable A random variable that has the following pdf is said to be a gamma random variable with parameters a, l > 0 l e- l x (l x)a 1 f ( x) (a ) 0 if x 0 if x 0 x where (a ) e x xa 1dx. 0 Note: for integer n, ( n) ( n 1)! The Normal random variable A random variable that has the following pdf is said to be a normal random variable with parameters m, s2 f ( x) 1 ( x m )2 / 2s 2 e 2s - x Note: The distribution with parameters m = 0 and s = 1 is called the standard normal distribution. Expectation of a random variable If X is a discrete random variable with pmf p(x), then the expected value of X is defined by E[ X ] xp( x) xP( X x) x x Expectation of a random variable If X is a discrete random variable with pmf p(x), then the expected value of X is defined by E[ X ] xp( x) xP( X x) x x Example: p(1)=0.2, p(3)=0.3, p(5)=0.2, p(7)=0.3 E[X] = 0.2(1)+0.3(3)+0.2(5)+0.3(7)=0.2+0.9+1+2.1=4.2 If X is a continuous random variable with pdf f(x), then the expected value of X is defined by E[ X ] xf ( x)dx Expectation of a Bernoulli random variable E[X] = 0(1 - p) + 1(p) = p Expectation of a Bernoulli random variable E[X] = 0(1 - p) + 1(p) = p Expectation of a geometric random variable 1 n 1 E[ X ] n 1 np(n) n 1 n(1 p) p p Expectation of a Bernoulli random variable E[X] = 0(1 - p) + 1(p) = p Expectation of a geometric random variable 1 n 1 E[ X ] n 1 np(n) n 1 n(1 p) p p Expectation of a binomial random variable n i n n E[ X ] i 0 ip(i ) i 0 i p (1 p) n i i Expectation of a Bernoulli random variable E[X] = 0(1 - p) + 1(p) = p Expectation of a geometric random variable 1 n 1 E[ X ] n 1 np(n) n 1 n(1 p) p p Expectation of a binomial random variable n i n n E[ X ] i 0 ip(i ) i 0 i p (1 p) n i i Expectation of a Poisson random variable E[ X ] i 0 ip(i ) i 0 ie l li i! l Expectation of a uniform random variable b x b2 a 2 a b E[ X ] dx a ba 2(b a) 2 Expectation of a uniform random variable b x b2 a 2 a b E[ X ] dx a ba 2(b a) 2 Expectation of an normal random variable 1 ( x m )2 / 2s 2 E[ X ] x e m 2s Expectation of a uniform random variable b x b2 a 2 a b E[ X ] dx a ba 2(b a) 2 Expectation of an exponential random variable 1 ( x m )2 / 2s 2 E[ X ] x e m 2s Expectation of a exponential random variable 1 l x E[ X ] xl e dx 0 l Expectation of a function of a random variable (1) If X is a discrete random variable with pmf p(x), then for any real-valued function g, E[ g ( X )] g ( x) p( x) g ( x) P ( X x) x x (2) If X is a continuous random variable with pdf f(x), then for any real-valued function g, E[ g ( X )] g ( x) f ( x)dx Expectation of a function of a random variable (1) If X is a discrete random variable with pmf p(x), then for any real-valued function g, E[ g ( X )] g ( x) p( x) g ( x) P ( X x) x x (2) If X is a continuous random variable with pdf f(x), then for any real-valued function g, E[ g ( X )] g ( x) f ( x)dx Note: P(Y=g(x))=P(X=x) If a and b are constants, then E[aX+b]=aE[X]+b The expected value E[Xn] is called the nth moment of the random variable X. The expected value E[(X-E[X])2] is called the variance of the random variable X and denoted by Var(X) Var(X) = E[X2] - E[X]2 Jointly distributed random variables Let X and Y be two random variables. The joint cumulative probability distribution of X and Y is defined as F (a, b) P{ X a, Y b}, - x FX (a ) P{ X a, Y } F (a, ) FY (a ) P{ X , Y b} F (, b) If X and Y are both discrete random variables, the joint pmf of X and Y is defined as p ( x, y ) P{ X x, Y y} p X ( x ) p ( x, y ) y pY ( y ) p ( x, y ) x If X and Y are continuous random variables, X and Y are said to be jointly continuous if there exists a function f(x, y) such that P( x A, y B) f ( x, y )dxdy A B P{ X A} P{ X A, Y (, )} P{ X A} f X ( x) fY ( y ) A f ( x, y )dxdy f ( x, y )dy f ( x, y )dx A f ( x, y )dxdy f X ( x)dx