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Lecture 7&8 notes

Lecture 7&8 notes - Chapter 5 PROBABILITY CONTINUED...

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Chapter 5 PROBABILITY CONTINUED: DISCRETE RANDOM VARIABLES 5.1 Discrete random variables Devore 7e: Sections 3.1, 3.2 Range, pmf, cdf 5.2 Expectation, mean and variance Devore 7e: Section 3.3 5.3 Several random variables Devore 7e: Pages 185–6 Joint pmf , marginal pmf , independence 5.4 Examples of discrete random variables Devore 7e: Sections 3.4–3.6 Discrete uniform, binomial, negative binomial, geometric, Poisson, hypergeometric 5.5 Chebyshev’s Inequality Devore 7e: Pages 107–8 (Exercise 44). 1

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5.1. RANDOM VARIABLES Random variable : A rule that assigns a real number to each outcome in the sam- ple space. See Figure 3.1 of Devore 7e for pictorial representation. Range R : set of possible values that the random variable can take on. Notation: We typically use upper case letters to denote random variables. We use lower case letters to denote a particular value that a random variable X takes on. For example, if basic outcome o occurs and X(o) = x , say, then we say the event { X = x } has occurred. Of course x must be a member of R . This is best understood through examples. Shorthand: event {X = x} = {o: X(o) = x} . Example: Toss 2 dice: X = sum { X = 4} = {(1,3), (2,2), (3,1)}. Discrete random variable : X only takes on discrete values. Example: X = No. of tosses till first head appears. R = {1,2,3,. . . }. Probability mass function ( pmf ) f(x) = P[X = x] (Note: Devore uses the notation p ( x ) rather than f ( x ) .) Properties f ( x ) 0 , x R f ( x ) = 1 Cumulative distribution function (cdf) F ( x ) = P [ X x ] = X x 0 x x 0 R f ( x 0 ) Increasing step function, F ( -∞ ) = 0 , F (+ ) = 1 2
Example: Geometric random variable Coin tossing: P (head) = p P (tail) = 1 - p = q. X = Number of tosses till first head f ( x ) = q x - 1 p, x = 1 , 2 , ...

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Lecture 7&8 notes - Chapter 5 PROBABILITY CONTINUED...

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