chap2 - Chapter 2 Discrete Random Variables 1 Basic...

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Unformatted text preview: Chapter 2 Discrete Random Variables 1 Basic Concepts • A random variable is a real-valued function of the outcome of the experiment. – The number of heads out of two tosses of a coin defines a function (mapping) from the sample space Ω = { HH, HT, T H, T T } into R : X ( HH ) = 2 , X ( HT ) = X ( T H ) = 1 and X ( T T ) = 0 . – The number of heads until the first tail in a sequence of tosses of a coin defines a function from the sample space Ω = { T, HT, HHT, HHHT, . . . } into R : X ( T ) = 0 , X ( HT ) = 1 , X ( HHT ) = 2 , X ( HHHT ) = 3 . . . • A function of a random variable defines another random variable. • A discrete random variable is a real-valued function of the outcome of the experiment that can take a finite or countably infinite number of values. • The event { ω : X ( ω ) ∈ A } is commonly denoted { X ∈ A } . 2 Probability Mass Functions • p X ( x ) = P [ X = x ] is the probability mass function of the random variable X . – Let X be the number of heads in the two tosses of a fair coin. X may take the values 0, 1 or 2: P [ X = 0] = 0 . 25 , P [ X = 1] = 0 . 50 and P [ X = 2] = 0 . 25 . – Let X be the number of heads until the first tail in a sequence of tosses of a coin: P [ X = 0] = 0 . 5 , P [ X = 1] = 0 . 25 , P [ X = 2] = 0 . 125 , P [ X = 3] = 0 . 0625 . . . Starting with $ y , you double your wealth after each head. Let Y be the amount of money you hold after the first tail: Y = 2 X y , P [ Y = y ] = 0 . 5 , P [ Y = 2 y ] = 0 . 25 , P [ Y = 4 y ] = 0 . 125 , P [ Y = 8 y ] = 0 . 0625 . . . 1 2 MTH2222 – Mathematics of Uncertainty • A probability mass functions must satisfy the following properties: – p ( x ) ≥ 0, for all x ’s in R – p ( x ) > 0, for at most a countable number of x ’s – X x p ( x ) = 1 • The Bernoulli Random Variable: A Bernoulli trial has only two possible outcomes usually referred to as success and failure. The Bernoulli random variable takes two values, 1 with probability say p , and 0 with probability 1- p : p ( x ) = 1- p x = 0 p x = 1 – A random variable with the above probability mass function is said to have a Bernoulli distribution with parameter p ....
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This note was uploaded on 10/09/2010 for the course MTH 2222 taught by Professor Kaizhamza during the Two '10 term at Monash.

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chap2 - Chapter 2 Discrete Random Variables 1 Basic...

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