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

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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

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

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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.

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Distribution of a random variable Distribution function 5 RANDOM VARIABLES

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

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

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Probability function 8 verifies RANDOM VARIABLES Distribution of a random variable

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

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

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

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Discrete uniform 12 1 2 3 k-1 k 1 2 3 k RANDOM VARIABLES Discrete random variables

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Bernoulli 13 0 1 p 1-p p RANDOM VARIABLES Discrete random variables

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

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

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

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

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

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

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

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

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(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

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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:

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

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

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

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Moment of order k 27 RANDOM VARIABLES

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Variance Given X with : standard deviation 28 RANDOM VARIABLES

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

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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)

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Correlation 31 RANDOM VARIABLES X and Y random variables;

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

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Expectation of transformed variables 33 RANDOM VARIABLES

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

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

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

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