2. Random variables  Introduction  Distribution of a random variable  Distribution function properties  Discrete random variables  Point mass  Discrete uniform  Bernoulli  Binomial  Geometric  Poisson 1

2. Random variables  Continuous random variables  Uniform  Exponential  Normal  Transformations of random variables  Bivariate random variables  Independent random variables  Conditional distributions  Expectation of a random variable  k th moment 2

2. Random variables  Variance  Covariance  Correlation  Expectation of transformed variables  Sample mean and sample variance  Conditional expectation 3

RANDOM VARIABLES Introduction Random variables assign a real number to each outcome: 4 Random variables can be:  Discrete: if it takes at most countably many values (integers).  Continuous: if it can take any real number.

Distribution of a random variable Distribution function 5 RANDOM VARIABLES

Distribution function properties 6 (i) when (ii) when (iii) is nondecreasing. (iv) is right-continuous. when RANDOM VARIABLES

7 For a random variable, we define Probability function Density function, depending on wether is either discrete or continuous Distribution of a random variable

Probability function 8 verifies RANDOM VARIABLES Distribution of a random variable

Probability density function 9 verifies We have RANDOM VARIABLES Distribution of a random variable

completely determines the distribution of a random variable. 10 RANDOM VARIABLES Distribution of a random variable

Discrete random variables Point mass 11 0 a 1-- RANDOM VARIABLES

Discrete uniform k-1 k k RANDOM VARIABLES Discrete random variables

Bernoulli p 1-p p RANDOM VARIABLES Discrete random variables

Binomial Successes in n independent Bernoulli trials with success probability p 14 RANDOM VARIABLES Discrete random variables

Geometric Time of first success in a sequence of independent Bernoulli trials with success probability p 15 RANDOM VARIABLES Discrete random variables

Poisson X expresses the number of “ rare events” 16 RANDOM VARIABLES Discrete random variables

Uniform 17 a b f(x) a b F(x) RANDOM VARIABLES Continuous random variables

Exponential 18 0  f(x) 1 F(x) 1/  RANDOM VARIABLES Continuous random variables

Normal 19  f(x) F(x) RANDOM VARIABLES Continuous random variables

Properties of normal distribution (i) standard normal (ii) (iii) independent i=1,2,...,n 20 RANDOM VARIABLES Continuous random variables

Transformations of random variables X random variable with ; Y = r(x); distribution of Y ? r() is one-to-one; r -1 (). 21 RANDOM VARIABLES

(X,Y) random variables;  If (X,Y) is a discrete random variable  If (X,Y) is continuous random variable 22 RANDOM VARIABLES Bivariate random variables

The marginal probability functions for X and Y are: 23 RANDOM VARIABLES Bivariate random variables For continuous random variables, the marginal densities for X and Y are:

Independent random variables Two random variables X and Y are independent if and only if: for all values x and y. 24 RANDOM VARIABLES

Conditional distributions Discrete variables 25 If X and Y are independent: Continuous variables RANDOM VARIABLES

Expectation of a random variable 26 Properties: (i) (ii)If are independent then: RANDOM VARIABLES

Moment of order k 27 RANDOM VARIABLES

Variance Given X with : standard deviation 28 RANDOM VARIABLES

Variance Properties: (i) (ii)If are independent then (iii) (iv) 29 RANDOM VARIABLES

Covariance X and Y random variables; 30 RANDOM VARIABLES Properties (i) If X, Y are independent then (ii) (iii) V(X + Y) = V(X) + V(Y) + 2cov(X,Y) V(X - Y) = V(X) + V(Y) - 2cov(X,Y)

Correlation 31 RANDOM VARIABLES X and Y random variables;

32 RANDOM VARIABLES Correlation Properties (i) (ii)If X and Y are independent then (iii)

Expectation of transformed variables 33 RANDOM VARIABLES

Sample mean and sample variance 34 Sample mean Sample variance RANDOM VARIABLES

Properties X random variable; i. i. d. sample, Then: (i) (ii) (iii) 35 RANDOM VARIABLES Sample mean and sample variance

Conditional expectation X and Y are random variables; Then: 36 Properties: RANDOM VARIABLES