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CHAPTER 4 4 4.1 - Discrete Models G eneral distributions C lassical: Binomial, Poisson, etc. 4 4.2 - Continuous Models G eneral distributions C lassical: Normal, etc.

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X Motivation ~ Motivation ~ Consider the following discrete random variable… 2 Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” Probability Table xf(x)f(x) 11/6 2 3 4 5 6 1 Probability Histogram “What is the probability of rolling a 4?” X is said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6. Total Area = 1 P(X = x) Density

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X 3 Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” Probability Table xf(x)f(x) 11/6 2 3 4 5 6 1 Probability Histogram “What is the probability of rolling a 4?” X is said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6. Total Area = 1 P(X = x) Motivation ~ Motivation ~ Consider the following discrete random variable… Density

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Motivation ~ Motivation ~ Consider the following discrete random variable… 4 Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” P(X = x) xf(x)f(x) 11/6 2 3 4 5 6 1 X is said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6. Cumulative distribution P(X x)P(X x) F(x)F(x) 1/6 2/6 3/6 4/6 5/6 1

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Motivation ~ Motivation ~ Consider the following discrete random variable… 5 Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” P(X = x) xf(x)f(x) 11/6 2 3 4 5 6 1 X is said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6. Cumulative distribution P(X x)P(X x) F(x)F(x) 1/6 2/6 3/6 4/6 5/6 1 “staircase graph” from 0 to 1

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Time intervals = 0.5 secsTime intervals = 2.0 secsTime intervals = 1.0 secs Time intervals = 5.0 secs “In the limit…” POPULATION random variable X 6 Example: X = “reaction time” “Pain Threshold” Experiment: Volunteers place one hand on metal plate carrying low electrical current; measure duration till hand withdrawn. “Pain Threshold” Experiment: Volunteers place one hand on metal plate carrying low electrical current; measure duration till hand withdrawn. In principle, as # individuals in samples increase without bound, the class interval widths can be made arbitrarily small, i.e, the scale at which X is measured can be made arbitrarily fine, since it is continuous. SAMPLE Total Area = 1 we obtain a density curve

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x 7 “In the limit…” x Cumulative probability F(x) = P(X x) = Area under density curve up to x f(x) no longer represents the probability P(X = x), as it did for discrete variables X. f(x) 0 Area = 1 f(x) = density function 00 x In fact, the zero area “limit” argument would seem to imply P(X = x) = 0 ???(Later…) However, F(x) increases continuously from 0 to 1. we can define “interval probabilities” of the form P( a X b ), using F(x). we obtain a density curve

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f(x) no longer represents the probability P(X = x), as it did for discrete variables X. 8 “In the limit…” Cumulative probability F(x) = P(X x) = Area under density curve up to x f(x) 0 Area = 1 f(x) = density function F(x) increases continuously from 0 to 1. a b a b However, we can define “interval probabilities” of the form P( a X b ), using F(x). F(a)F(a) F(b)F(b) F( b ) F( a ) we obtain a density curve In fact, the zero area “limit” argument would seem to imply P(X = x) = 0 ???(Later…)

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An “interval probability” P( a X b ) can be calculated as the amount of area under the curve f(x) between a and b, or the difference P(X b ) P(X a ), i.e., F( b ) F( a ). (Ordinarily, finding the area under a general curve requires calculus techniques… unless the “curve” is a straight line, for instance. Examples to follow…) f(x) no longer represents the probability P(X = x), as it did for discrete variables X. 9 “In the limit…” Cumulative probability F(x) = P(X x) = Area under density curve up to x f(x) 0 Area = 1 f(x) = density function a b a b F(x) increases continuously from 0 to 1. F(a)F(a) F(b)F(b) F( b ) F( a ) we obtain a density curve

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Moreover, and. 10 f(x) = density function Cumulative probability F(x) = P(X x) = Area under density curve up to x Thus, in general, P( a X b ) = = F( b ) F( a ). “In the limit…” f(x) 0 Area = 1 F(x) increases continuously from 0 to 1. Fundamental Theorem of Calculus we obtain a density curve

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X Consider the following continuous random variable… 11 Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” “What is the probability of rolling a 4?” Probability Histogram Total Area = 1 Probability Table xf(x)f(x) 11/6 2 3 4 5 6 1 P(X = x) F(x)F(x) 1/6 2/6 3/6 4/6 5/6 1 Cumul Prob P(X x) “staircase graph” from 0 to 1 Density

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X Consider the following continuous random variable… 12 Example: X = “Ages of children from 1 year old to 6 years old” “What is the probability of rolling a 4?” Further suppose that X is uniformly distributed over the interval [1, 6]. Probability Histogram Total Area = 1 Probability Table xf(x)f(x) 11/6 2 3 4 5 6 1 P(X = x) F(x)F(x) 1/6 2/6 3/6 4/6 5/6 1 Cumul Prob P(X x) “staircase graph” from 0 to 1 Density

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F(x)F(x) 1/6 2/6 3/6 4/6 5/6 1 Probability Table xf(x)f(x) 11/6 2 3 4 5 6 1 X Consider the following continuous random variable… 13 Example: X = “Ages of children from 1 year old to 6 years old” “What is the probability of rolling a 4?” Further suppose that X is uniformly distributed over the interval [1, 6]. Probability Histogram Total Area = 1 P(X = x) Cumul Prob P(X x) “staircase graph” from 0 to 1 Density

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X Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” “What is the probability of rolling a 4?” Further suppose that X is uniformly distributed over the interval [1, 6]. Total Area = 1 Cumul Prob P(X x) that a random child is 4 years old?” Check? Base = 6 – 1 = 5 Height = 0.2 5 0.2 = 1 doesn’t mean….. > 0 The probability that a continuous random variable is exactly equal to any single value is ZERO! Density A single value is one point out of an infinite continuum of points on the real number line. F(x)F(x)

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F(x)F(x) X Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” “What is the probability of rolling a 4?” Further suppose that X is uniformly distributed over the interval [1, 6]. Cumul Prob P(X x) that a random child is 4 years old?”actually means.... = (5 – 4)(0.2) = 0.2 between 4 and 5 years old?” < << NOTE: Since P(X = 5) = 0, no change for P(4 X 5), P(4 < X 5), or P(4 < X < 5). Density Alternate way using cumulative distribution function (cdf)…

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F(x)F(x) X Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” “What is the probability of rolling a 4?” Further suppose that X is uniformly distributed over the interval [1, 6]. Cumul Prob P(X x) that a random child is under 5 years old? Density 0.8 Alternate way using cumulative distribution function (cdf)…

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F(x)F(x) X Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” “What is the probability of rolling a 4?” Further suppose that X is uniformly distributed over the interval [1, 6]. Cumul Prob P(X x) that a random child is under 4 years old? Density 0.6 Alternate way using cumulative distribution function (cdf)…

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F(x)F(x) X Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” “What is the probability of rolling a 4?” Further suppose that X is uniformly distributed over the interval [1, 6]. Cumul Prob P(X x) that a random child is Density between 4 and 5 years old?” Alternate way using cumulative distribution function (cdf)…

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F(x)F(x) X Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” “What is the probability of rolling a 4?” Further suppose that X is uniformly distributed over the interval [1, 6]. Cumul Prob P(X x) that a random child is Density between 4 and 5 years old?” = F(5) F(4) Alternate way using cumulative distribution function (cdf)… = 0.8 – 0.6 = 0.2

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F(x)F(x) X Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose that X is uniformly distributed over the interval [1, 6]. Cumul Prob P(X x) Cumulative probability F(x) = P(X x) = Area under density curve up to x x For any x, the area under the curve is F(x) = 0.2 (x – 1). Density

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Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose that X is uniformly distributed over the interval [1, 6]. x For any x, the area under the curve is F(x) = 0.2 (x – 1). Density Cumulative probability F(x) = P(X x) = Area under density curve up to x F(x) = 0.2 (x – 1) F(x) increases continuously from 0 to 1. (compare with “staircase graph” for discrete case)

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Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose that X is uniformly distributed over the interval [1, 6]. Density Cumulative probability F(x) = P(X x) = Area under density curve up to x F(x) = 0.2 (x – 1) F(4) = 0.6 F(5) = 0.8 “What is the probability that a child is between 4 and 5?” = F(5) F(4) = 0.8 – 0.6 = 0.2 0.2

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Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose that X is uniformly distributed over the interval [1, 6]. Density > 0 Area = Base Height = 1

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Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” “What is the probability that a child is under 4 years old?” Density Area = Base Height

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Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Density Area = Base = 0.36 Alternate method, without having to use f(x): Use proportions via similar triangles. h = ? 0.36 “What is the probability that a child is under 4 years old?”

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Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” “What is the probability that a child is under 4 years old?” Density “What is the probability that a child is over 4 years old?” 0.36 0.64

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Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Cumulative probability F(x) = P(X P(X x)x) = Area under density curve up to x F(x) = ???????? “What is the probability that a child is under 4 years old?” Exercise… Density x “What is the probability that a child is under 5 years old?” “What is the probability that a child is between 4 and 5?”

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Unfortunately, the cumulative area (i.e., probability) under most curves either… requires “integral calculus,” or is numerically approximated and tabulated. 28 IMPORTANT SPECIAL CASE: “Bell Curve”

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