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EMSE 208 Lecture 2 - Random Variables

# EMSE 208 Lecture 2 - Random Variables - 1 Lecture 2 Random...

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1 Lecture 2: Random Variables Professor Thomas A. Mazzuchi

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2 Random Variable Definition A random variable X, is a real valued function on a probability space X:S →ℜ such that for all t in we can assign a probability to the event X t. We use capital letters to denote random variables and small letters to denote their value S * * * * * * 0 t a 4 a 5 a 6 a 3 a 1 a 2 P(X t) = P(a 2 )+P(a 5 )+P(a 6 )
3 Random Variable Example: Experiment - Tossing a coin until a heads appears where P(H) = p S = { H, TH, TTH, …. .} Let N = # of tosses Pr{N=n} = (1-p) n-1 p for n=1,2,… Note that the probability that N takes on any value from 1 to is equivalent to the sample space 1 1 1 1 1 1 0 { } ( ) (1 ) = ) ) 1 = 1 1 (1 ) n n n n n n n n P N n P N n p p p p p p p p - = = = - = = = = = = - - = - = - - U

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4 Random Variable Example: Indicator (Bernoulli) Random Variables – Suppose we consider the life of a battery, T S= [0, ) P(i) = (1- P(T>t)) 1-i (P(T>t)) i i=0,1 1 if battery life t 0 otherwise I =
5 Representing Random Variable Probability Mass Function (pmf) When variables take on values that are countable or listable from smallest to largest they are called Discrete Random Variables A pmf, f(x), of a random variable X is a function representing the values of the random variable with their associated probabilities. This for a pmf f(x) = P(X=x) and 0 f(x) 1 Σ x f(x)=1 As it is a function, it can be specified in tabular, graphic or equational form X f(x) 0 0.16 f(x) 1 0.48 2 0.36 0 1 2 x x x x f - = 2 ) 4 . 0 ( ) 6 . 0 ( 2 ) ( x

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6 Representing Random Variable Cumulative Distribution Function (CDF) A CDF, F(x), of a random variable X is a function representing P(X x), thus 0 F(x) 1 Additional Properties for the CDF F(x) is a nondecreasing function of x Lim x →∞ F(x) = 1 Lim x - F(x) = 0 Using the CDF to Calculate Probabilities P( a < X b} =F(b) - F(a) P( a < X < b} =F(b - ) - F(a) P( a X < b} =F(b - ) - F(a - ) P( a X b} =F(b) - F(a - ) - -
7 Representing Random Variable Cumulative Distribution Function (CDF) for Discrete RVs For discrete random variables, this is a step function 0 5 10 15 1.0 0.8 0.6 0.4 0.2 f(x) x 0 5 10 15 0.6 0.4 0.2 x Note that the size of the jump in F(x*) is the value of f(x*) Note for example P(X 12) = F(12) = 0.8

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8 Special Discrete Random Variables There are some variables that have defined functional forms and are dependent only on one or two parameters. The pmf’s of these random variables are said to belong to a Parametric Family. Identifying these variables is important as we may reduce the amount of modeling and estimation effort.
9 Special Discrete Random Variables 1 pmf ( | ) (1 ) 0,1, x x f x p p p x - = - = Bernoulli Random Variable Outcome of a trial is classified one of two ways we arbitrarily call one of those ways a success and the other a failure P(success) = p Let X be the number of successes Uniform Random Variable Consider an n equally likely outcomes x 1 , …, x n n ,...,x x x n n x f 1 for 1 ) | ( pmf = =

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10 Special Discrete Random Variables Binomial Random Variable
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EMSE 208 Lecture 2 - Random Variables - 1 Lecture 2 Random...

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