Chapter 8 Lecture Outline

Chapter 8 Lecture Outline - CHAPTER 8 Random Variables -...

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CHAPTER 8 Random Variables - assigns a number (or symbol) to each outcome of a random circumstance. Examples: Call our random variable X! Let X = the number of spades in a random sample of 4 cards from a deck. Let X = the sum of 2 rolls of a six sided die. Let X = the number of people with blue eyes in a sample of 10 people. Let X = the weight of a randomly chosen person. Discrete Random Variables – a variable that can only result in a countable set of possibilities. This implies that a discrete random variable cannot take any possible value in an interval (since there are uncountable many values in an interval). Examples: 1. X = the sum of 2 rolls of a six sided die. Outcomes: 2-12 => Discrete 2. X = the weight of a randomly chosen individual. Outcomes: 120 lbs, 120.00001lbs, 183.12302 lbs,… => NOT Discrete 3. X = number of tosses until the first “head”. Outcomes: 1,2,3,… => Discrete

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Probability Notation and Probability Distributions X = the random variable. k = a number that the discrete r.v. could assume (a possible outcome!) P(X=k) is the probability that X equals k. The probability distribution function (PDF) for a discrete random variable is a table or rule that assigns probabilities to the possible outcomes of a random variable X. Example Assume the probability of a girl is ½. Let X = the number of girls in a family with 3 children. What is the probability distribution of X? Possible Outcomes: 0, 1, 2, or 3 girls. Event: BBB BBG BGB GBB BG G GB G GG B GG G Prob: 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8 X: 0 1 1 1 2 2 2 3 PDF of X: k P(X=k) 0 1/8 1 3/8 2 3/8
3 1/8 Cumulative Distribution Function The Cumulative Distribution Function for a random variable X is a rule or table that provides the probabilities P(X ≤ k). The term ‘cumulative probability’ refers to the probability that X is less than or equal to a particular value. Example: Number of Girls CDF of X: k P(X≤k) 0 1/8 1 1/8+3/8 = 1/2 2 1/8+3/8+3/8 = 7/8 3 everything = 1 Expectations Expected value of a random variable - the mean value of the variable X in the space of possible outcomes (or population). Also can be interpreted as the mean value from an infinite number of observations of the random variable. It is also denoted with the Greek letter μ (pronounced mu).

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This note was uploaded on 03/30/2008 for the course L I R 201 taught by Professor Willits,billie during the Spring '07 term at Penn State.

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Chapter 8 Lecture Outline - CHAPTER 8 Random Variables -...

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