# chapter6 - CHAPTER 6 PROBABILITY DISTRIBUTIONS 6.1 Random...

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CHAPTER 6: PROBABILITY DISTRIBUTIONS 6.1 Random Variables Random variable assigns a numerical value to each outcome of a random experiment. Random Variable Possible values X= Number of defectives in a random sample of two computer chips drawn from a shipment X=0, 1, 2 X= the waiting time until the first customer arrives at a checkout counter (the counter will be open for 8 hours). [ ] 0,8 X The probability distribution of a random variable X tells us what the possible values of X are and how probabilities are assigned to those values. Types of Random Variables: Discrete: possible values are isolated points on the number line (number of defectives in a sample). Continuous : possible values lie in an interval (waiting time). 1

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6.2 Discrete Random Variables Probability distribution of a discrete random variable is specified by a table: x x 1 x 2 ... x n P(X=x) p 1 p 2 ... p n Here x i are the possible values of X and p i the corresponding probabilities (sum of p i values equal to 1). Example: 10% of computer chips in a very large shipment are defective. Denote by X the number of defective chips in a random sample of two. Find the probability distribution of X. What is P(X 1)? Solution: Probability histogram: 0.81 0.18 0.01 0 1 2 2
Mean of a discrete probability distribution Example: Average age of children Age 1 2 3 4 5 6 7 8 9 Fraction .01 .03 .05 .07 .17 .25 .20 .15 .07 Average Age =? The mean or expected value of a discrete random variable X, denoted as μ is () x xPX x μ =⋅ = , where the sum is over all possible values x of X. Example: Calculate the mean of X=the number of defectives in a random sample of two. 3

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Example: Suppose that someone offers you bet of flipping two fair coins. If two heads are obtained, you win \$10, otherwise you must pay \$6. What are your expected winnings? 6.3 Continous Random Variables Probability distribution of a continuous random variable is specified by a density curve: a b P(a X b) P(a X b) = the probability that the values of X are between a and b = Area under the density curve between a and b. 4
Example: X = Weight of women in a population Relative Freq. 130 Total area of all rectangles = 1 Area=fraction of women of weight between 129 and 130 129 X Relative Freq. 130 Area=fraction of women of weight smaller than 130 129 X Relative distribution of X approximated by a smooth curve ( density curve ) Freq. 5

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Relative Freq. ••••••• Density curve describes the overall pattern of a distribution Total area under the curve = 1 Relative Freq.
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chapter6 - CHAPTER 6 PROBABILITY DISTRIBUTIONS 6.1 Random...

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