Chapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions § 4. 1 - Probability Density Functions § 4. 2 - Cumulative Distribution Functions and Expected Values § 4. 3 - The Normal Distribution § 4. 4 - The Exponential and Gamma Distributions § 4. 5 - Other Continuous Distributions § 4. 6 - Probability Plots
POPULATION Discrete random variable X “Density” Pop vals pmf x p (x ) x 1 p(x 1) x 2 p(x 2) x 3 p(x 3) ⋮ ⋮ Total 1 Total Area = 1 p(x) = Probability that the random variable X is equal to a specific value x, i. e. , p(x) = P(X = x) “probability mass function” (pmf) | x X
POPULATION Discrete random variable X Pop vals pmf x p (x ) x 1 p(x 1) F(x 1) = p(x 1) x 2 p(x 2) F(x 2) = p(x 1) + p(x 2) x 3 p(x 3) F(x 3) = p(x 1) + p(x 2) + p(x 3) ⋮ ⋮ ⋮ Total 1 increases from 0 to 1 cdf F (x ) = P (X x ) Total Area = 1 “staircase graph” F(x) = Probability that the random variable X is less than or equal to a specific value x, i. e. , F(x) = P(X x) “cumulative distribution function” (cdf) | x X
POPULATION Discrete random variable X Pop vals pmf cdf x p (x ) x 1 p(x 1) F(x 1) = p(x 1) x 2 p(x 2) F(x 2) = p(x 1) + p(x 2) x 3 p(x 3) F(x 3) = p(x 1) + p(x 2) + p(x 3) ⋮ ⋮ ⋮ Total 1 increases from 0 to 1 F (x ) = P (X x ) Calculating “interval probabilities”… F(b) = P(X b) F(a–) = P(X a–) F(b) – F(a–) = P(X b) – P(X a–) = P(a X b) p(x) | | a–a | b X
POPULATION Discrete random variable X Pop vals pmf cdf x p (x ) x 1 p(x 1) F(x 1) = p(x 1) x 2 p(x 2) F(x 2) = p(x 1) + p(x 2) x 3 p(x 3) F(x 3) = p(x 1) + p(x 2) + p(x 3) ⋮ ⋮ ⋮ Total 1 increases from 0 to 1 F (x ) = P (X x ) Calculating “interval probabilities”… F(b) = P(X b) F(a–) = P(X a–) F(b) – F(a–) = P(X b) – P(X a–) = P(a X b) p(x) | | a–a | b X FUNDAMENTAL THEOREM OF CALCULUS (discrete form)
Reconsider the following discrete random variable… Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” X is uniformly distributed over 1, 2, 3, 4, 5, 6. Probability Table Cumul Probability Histogram Density Total Area = 1 X “What is the probability of rolling a 4? ” P(X = x) P(X x) x p(x) F(x) 1 1/6 2/6 3 1/6 3/6 4 1/6 4/6 5 1/6 5/6 6 1/6 1 1 6
Reconsider the following discrete random variable… Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” X is uniformly distributed over 1, 2, 3, 4, 5, 6. Probability Table Cumul Probability Histogram Density Total Area = 1 Not a histogram! “What is the probability of rolling a 4? ” “staircase graph” from 0 to 1 X P(X = x) P(X x) x p(x) F(x) 1 1/6 2/6 3 1/6 3/6 4 1/6 4/6 5 1/6 5/6 6 1/6 1 1 7
Reconsider Consider thethe following continuous discrete randomvariable… Example: X X= children 1 year old to 6 years old”die (1, 2, 3, 4, 5, 6)” Example: = “Ages “value of shown on afrom single random toss of a fair 3, 4, 5, [1, 6. 6]. Further suppose that X is uniformly distributed over the 2, interval over 1, Probability Table Cumul Probability Histogram Density Total Area = 1 X “What is the probability of rolling a 4? ” P(X = x) P(X x) x p(x) F(x) 1 1/6 2/6 3 1/6 3/6 4 1/6 4/6 5 1/6 5/6 6 1/6 1 1 8
Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” 3, 4, 5, [1, 6. 6]. Further suppose that X is uniformly distributed over the 2, interval over 1, Probability Table Cumul Probability Histogram Density Total Area = 1 X “What is the probability of rolling a 4? ” P(X = x) P(X x) x f(x) F(x) 1 1/6 2/6 3 1/6 3/6 4 1/6 4/6 5 1/6 5/6 6 1/6 1 1 9
Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” 3, 4, 5, [1, 6. 6]. Further suppose that X is uniformly distributed over the 2, interval over 1, Probability Table Cumul Probability Histogram Density Total Area = 1 X “What is the probability of a child rolling is a 4 4? ” years old? ” P(X = x) P(X x) x f(x) F(x) 1 1/6 2/6 3 1/6 3/6 4 1/6 4/6 5 1/6 5/6 6 1/6 1 1 10
POPULATION Discrete random variable X Continuous Example: X = “reaction time” “Pain Threshold” Experiment: Volunteers place one hand on metal plate carrying low electrical current; measure duration till hand withdrawn. 0. 5 Time intervals = 5. 0 2. 0 secs Time intervals = 1. 0 secs “In the limit…” we obtain a density curve Total Area = 1 SAMPLE 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. 11
“In the limit…” we obtain a density curve Cumulative probability F(x) = P(X x) = Area under density curve up to x f(x) = probability density function (pdf) • f(x) 0 • Area = 1 00 F(x) increases continuously from 0 to 1. x x x As with discrete variables, the density f(x) is the height, NOT the probability p(x) = P(X = 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 cdf F(x). 12
“In the limit…” we obtain a density curve Cumulative probability F(x) = P(X x) = Area under density curve up to x F(b) f(x) = probability density function (pdf) F(b) F(a) • f(x) 0 • Area = 1 a b F(x) increases continuously from 0 to 1. a b As with discrete variables, the density f(x) is the height, NOT the probability p(x) = P(X = 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 cdf F(x). 13
“In the limit…” we obtain a density curve Cumulative probability F(x) = P(X x) = Area under density curve up to x F(b) f(x) = probability density function (pdf) F(b) F(a) • f(x) 0 • Area = 1 a b F(x) increases continuously from 0 to 1. a b 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…) 14
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]. >0 Density Total Area = 1 Check? Base = 6 – 1 = 5 5 0. 2 = 1 Height = 0. 2 X “What is the probability of that rolling a random a 4? ” child is 4 years old? ” doesn’t mean…. . = 0 !!!!! A single value is one point out of an infinite continuum of points on the real number line. The probability that a continuous random variable is exactly equal to any single value is ZERO!
Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Density Further suppose that X is uniformly distributed over the interval [1, 6]. X “What is the probability of rolling a 4? ” child is 4 between 4 and 5 years old? ” that a random 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).
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]. Cumulative probability F(x) = P(X x) = Area under density curve up to x Density For any x, the area under the curve is F(x) = 0. 2 (x – 1). 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]. Cumulative probability F(x) = P(X x) = Area under density curve up to x F(x) = 0. 2 (x – 1) Density For any x, the area under the curve is F(x) increases continuously from 0 to 1. F(x) = 0. 2 (x – 1). (compare with “staircase graph” for discrete case) 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]. Cumulative probability F(x) = P(X x) = Area under density curve up to x F(x) = 0. 2 (x – 1) Density F(5) = 0. 8 X “What is the probability of rolling a 4? ” child is under 5 years old? that a random 0. 8
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]. Cumulative probability F(x) = P(X x) = Area under density curve up to x Density F(x) = 0. 2 (x – 1) F(4) = 0. 6 X “What is the probability of rolling a 4? ” child is under 4 years old? that a random 0. 6
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]. Cumulative probability F(x) = P(X x) = Area under density curve up to x F(x) = 0. 2 (x – 1) Density F(5) = 0. 8 F(4) = 0. 6 X “What is the probability of rolling a 4? ” child is between 4 and 5 years old? ” that a random
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]. Cumulative probability F(x) = P(X x) = Area under density curve up to x F(x) = 0. 2 (x – 1) Density F(5) = 0. 8 0. 2 F(4) = 0. 6 X “What is the probability of rolling a 4? ” child is between 4 and 5 years old? ” that a random = F(5) F(4) = 0. 8 – 0. 6 = 0. 2
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]. 0 Density Area = Base Height =1
Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Density Cumulative Distribution Function F(x) Cumulative probability F(x) = P(X x) = Area under density curve up to x x x
Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Density Cumulative Distribution Function F(x) Cumulative probability F(x) = P(X x) = Area under density curve up to x 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? ”
A continuous random variable X Cumulative probability function (cdf) In summary… corresponds to a probability density function (pdf) f(x), whose graph is a density curve. f(x) is NOT a pmf! Fundamental Theorem of Calculus Moreover… F(x) increases continuously and monotonically from 0 to 1. 26
A continuous random variable X Cumulative probability function (cdf) In summary… corresponds to a probability density function (pdf) f(x), whose graph is a density curve. f(x) is NOT a pmf! Fundamental Theorem of Calculus Moreover… F(x) increases continuously and monotonically from 0 to 1. 27
SECTION 4. 3 IN POSTED LECTURE NOTES
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Four Examples: 1 For any b > 0, consider the following probability density function (pdf). . . Confirm pdf: Is clear by inspection. � �
Four Examples: 1 For any b > 0, consider the following probability density function (pdf). . . Determine the cumulative distribution function (cdf) For any x < 0, it follows that… For any it follows that…
Four Examples: 1 For any b > 0, consider the following probability density function (pdf). . . Determine the cumulative distribution function (cdf) For any x < 0, it follows that For any it follows that…
Four Examples: 1 � For any b > 0, consider the following probability density function (pdf). . . Determine the cumulative distribution function (cdf) For any x < 0, it follows that For any it follows that… Note: For any it follows that…
Four Examples: 1 � For any b > 0, consider the following probability density function (pdf). . . Determine the cumulative distribution function (cdf) Monotonic and continuous from 0 to 1 34
Four Examples: 2 Consider the following probability density function (pdf). . . Confirm pdf: Exercise 35
Four Examples: 2 Consider the following probability density function (pdf). . . Determine the cumulative distrib function (cdf) For any it follows that 36
Four Examples: 2 Consider the following probability density function (pdf). . . Determine the cumulative distrib function (cdf) For any it follows that 37
Four Examples: 2 Consider the following probability density function (pdf). . . Determine the cumulative distrib function (cdf) For any it follows that… For any it follows that 38
Four Examples: 2 Consider the following probability density function (pdf). . . Determine the cumulative distrib function (cdf) 39
Four Examples: 2 Consider the following probability density function (pdf). . . cumulative distrib function (cdf) Determine the mean Determine the variance Determine the median ? ? ? ? 40
Four Examples: 2 Consider the following probability density function (pdf). . . Determine the cumulative distrib function (cdf) 41
Four Examples: 3 Consider the following probability density function (pdf). . . Confirm pdf WARNING: “IMPROPER INTEGRAL” �
Four Examples: 4 3 Consider the following probability density function (pdf). . . Confirm pdf WARNING: “IMPROPER INTEGRAL” �
Four Examples: 4 3 Consider the following probability density function (pdf). . . Confirm pdf WARNING: “IMPROPER INTEGRAL” � does not exist!
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