04 Discrete random variables

04 Discrete random variables - University of California,...

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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Random variables • Discrete random variables. • Continuous random variables. • Discrete random variables . Denote a discrete random variable with X : It is a variable that takes values with some probability. Examples: a. Roll a die. Let X be the number observed. b. Draw 2 cards with replacement. Let X be the number of aces among the 2 cards. c. Roll 2 dice. Let X be the sum of the 2 numbers observed. d. Toss a coin 5 times. Let X be the number of tails among the 5 tosses. e. Randomly select a US household. Let X be the number of people live in this household. • Probability distribution of a discrete random variable X It is the list of all possible values of X with the corresponding probabilities. It can be represented by a table, a graph, or a function. Examples: a. Roll a die. Let X be the number observed. The probability distribution of X is: X P ( X = x ) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6 X P(X=x) 1 2 3 4 5 6 0.00 0.05 0.10 0.15 1 b. Roll two dice. Let X be the sum of the two numbers observed. The probability distribution of X is: X P ( X = x ) 2 1 36 3 2 36 4 3 36 5 4 36 6 5 36 7 6 36 8 5 36 9 4 36 10 3 36 11 2 36 12 1 36 X P(X=x) 2 3 4 5 6 7 8 9 10 11 12 0.00 0.05 0.10 0.15 We can also represent this distribution with a function: P ( X = x ) = 6-| x- 7 | 36 , for x = 2 , 3 , ··· , 12. This is called probability mass function and returns the probability for each possible value of the random variable X . • Expected value (or mean) of a discrete random variable It is denoted with E ( X ) or μ and it is computed as follows: Definition: μ = E ( X ) = X x xP ( X = x ) It is a weighted average. The weights are the probabilities....
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This note was uploaded on 01/14/2011 for the course STATS 100A taught by Professor Wu during the Fall '07 term at UCLA.

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04 Discrete random variables - University of California,...

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