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CHAPTER 4

CHAPTER 4. 4.1 - Discrete Models General distributions Classical: Binomial, Poisson, etc. 4.2 - Continuous Models General distributions Classical: Normal, etc. Motivation ~ Consider the following discrete random variable….

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CHAPTER 4

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  1. CHAPTER 4 4.1 - Discrete Models General distributions Classical: Binomial, Poisson, etc. 4.2 - Continuous Models General distributions Classical: Normal, etc.

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

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

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

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

  6. POPULATION Time intervals = 0.5 secs Example: X= “reaction time” “Pain Threshold” Experiment: Volunteers place one hand on metal plate carrying low electrical current; measure duration till hand withdrawn. Time intervals = 1.0 secs “In the limit…” Continuous random variable X we obtain a density curve Time intervals = 2.0 secs Total Area = 1 Time intervals = 5.0 secs SAMPLE Time intervals = 1.0 secs 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.

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 x x x f(x) no longer represents the probability P(X = x), as it did for discrete variables X. In fact, the zero area “limit” argument would seem to imply P(X = x) = 0 ???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

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

  9. Cumulative probability F(x) = P(X x) = Area under density curve up to x “In the limit…” we obtain a density curve F(b) f(x) = density function F(b) F(a) F(a) • f(x)  0 • Area = 1 F(x) increases continuously from 0 to 1. b a b a f(x) no longer represents the probability P(X = x), as it did for discrete variables X. An “interval probability” P(a X b) can be calculated as the amount of area under the curve f(x) betweena 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…)

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

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

  12. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. Probability Histogram CumulProb P(Xx) P(X = x) Total Area = 1 Density X “What is the probability of rolling a 4?” “staircase graph” from 0 to 1

  13. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. Probability Histogram CumulProb P(Xx) P(X = x) Total Area = 1 Density X “What is the probability of rolling a 4?” “staircase graph” from 0 to 1

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

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

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

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

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

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

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

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

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

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

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

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

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

  27. 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 x) = Area under density curve up to x Exercise… F(x) = ???????? ? Density x “What is the probability that a child is under 4 years old?” “What is the probability that a child is under 5 years old?” “What is the probability that a child is between 4 and 5?”

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

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