x1 p(x1) x2 p(x2) x3 p(x3) POPULATION x p(x) ⋮ Total 1 “Density” Example: X = Cholesterol level (mg/dL) random variable X Discrete Pop vals x pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 “Density” Total Area = 1 X 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

x1 p(x1) x2 p(x2) x3 p(x3) x1 p(x1) x2 p(x2) x3 p(x3) POPULATION x Example: X = Cholesterol level (mg/dL) random variable X Discrete Pop vals x pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Pop vals x pmf p(x) cdf F(x) = P(X  x) x1 p(x1) F(x1) = p(x1) x2 p(x2) F(x2) = p(x1) + p(x2) x3 p(x3) F(x3) = p(x1) + p(x2) + p(x3) ⋮ Total 1 increases from 0 to 1 Total Area = 1 “staircase graph” X 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

Example: X = Cholesterol level (mg/dL) POPULATION Example: X = Cholesterol level (mg/dL) random variable X Discrete Pop vals x pmf p(x) cdf F(x) = P(X  x) x1 p(x1) F(x1) = p(x1) x2 p(x2) F(x2) = p(x1) + p(x2) x3 p(x3) F(x3) = p(x1) + p(x2) + p(x3) ⋮ Total 1 increases from 0 to 1 Calculating “interval probabilities”… X 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

FUNDAMENTAL THEOREM OF CALCULUS POPULATION Pop vals x pmf p(x) cdf F(x) = P(X  x) x1 p(x1) F(x1) = p(x1) x2 p(x2) F(x2) = p(x1) + p(x2) x3 p(x3) F(x3) = p(x1) + p(x2) + p(x3) ⋮ Total 1 increases from 0 to 1 Discrete random variable X Example: X = Cholesterol level (mg/dL) Calculating “interval probabilities”… X F(b) = P(X  b) F(a–) = P(X  a–) F(b) – F(a–) = P(X  b) – P(X  a–) FUNDAMENTAL THEOREM OF CALCULUS (discrete form) = P(a  X  b) p(x) | a– | a | b

Probability Histogram 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 x p(x) 1 1/6 2 3 4 5 6 Probability Histogram Cumul Prob P(X  x) P(X = x) X F(x) 1/6 2/6 3/6 4/6 5/6 1 Total Area = 1 Density “What is the probability of rolling a 4?”

Probability Histogram “staircase graph” from 0 to 1 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 x p(x) 1 1/6 2 3 4 5 6 Probability Histogram Cumul Prob P(X  x) P(X = x) X F(x) 1/6 2/6 3/6 4/6 5/6 1 Total Area = 1 Density “What is the probability of rolling a 4?” “staircase graph” from 0 to 1

Consider the following continuous random variable… 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)” 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 is uniformly distributed over 1, 2, 3, 4, 5, 6. Probability Table x p(x) 1 1/6 2 3 4 5 6 Probability Histogram Cumul Prob P(X  x) P(X = x) X F(x) 1/6 2/6 3/6 4/6 5/6 1 Total Area = 1 Density “What is the probability of rolling a 4?”

Probability Histogram 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 is uniformly distributed over 1, 2, 3, 4, 5, 6. Probability Table x f(x) 1 1/6 2 3 4 5 6 Probability Histogram Cumul Prob P(X  x) P(X = x) X F(x) 1/6 2/6 3/6 4/6 5/6 1 Total Area = 1 Density “What is the probability a child is 4 years old?” “What is the probability of rolling a 4?”

POPULATION Continuous Discrete random variable X “In the limit…” Time intervals = 0.5 secs Time intervals = 5.0 secs Time intervals = 2.0 secs Time intervals = 1.0 secs Example: X = Cholesterol level (mg/dL) Example: X = “reaction time” “Pain Threshold” Experiment: Volunteers place one hand on metal plate carrying low electrical current; measure duration till hand withdrawn. 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.

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) = probability density function (pdf) f(x)  0 Area = 1 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).

F(b) F(b)  F(a) F(a) However, 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) = probability density function (pdf) F(b)  F(a) F(a) f(x)  0 Area = 1 F(x) increases continuously from 0 to 1. a b 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).

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) = probability density function (pdf) F(b)  F(a) F(a) f(x)  0 Area = 1 F(x) increases continuously from 0 to 1. a b 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…)

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  X Total Area = 1 Density Check? Base = 6 – 1 = 5 5  0.2 = 1  Height = 0.2 “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 exactly equal to any single value is ZERO! A single value is one point out of an infinite continuum of points on the real number line.

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 Density “What is the probability of rolling a 4?” that a random child is 4 years old?” between 4 and 5 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 X For any x, the area under the curve is Density F(x) = 0.2 (x – 1). 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) 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 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 X F(x) = 0.2 (x – 1) F(5) = 0.8 Density “What is the probability of rolling a 4?” that a random child is under 5 years old? 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 X F(x) = 0.2 (x – 1) Density F(4) = 0.6 “What is the probability of rolling a 4?” that a random child is under 4 years old? 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 X F(x) = 0.2 (x – 1) F(5) = 0.8 Density F(4) = 0.6 “What is the probability of rolling a 4?” that a random child is between 4 and 5 years old?”

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 X F(x) = 0.2 (x – 1) F(5) = 0.8 0.2 Density F(4) = 0.6 “What is the probability of rolling a 4?” that a random child is between 4 and 5 years old?” = 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  Area = Base  Height = 1  Density

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 Cumulative Distribution Function F(x) Density x x

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 Cumulative Distribution Function 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?”

Fundamental Theorem of Calculus A continuous random variable X corresponds to a probability density function (pdf) f(x), whose graph is a density curve. f(x) is NOT a pmf! Cumulative probability function (cdf) In summary… Fundamental Theorem of Calculus F(x) increases continuously from 0 to 1. Moreover…

Fundamental Theorem of Calculus A continuous random variable X corresponds to a probability density function (pdf) f(x), whose graph is a density curve. f(x) is NOT a pmf! Cumulative probability function (cdf) In summary… Fundamental Theorem of Calculus F(x) increases continuously from 0 to 1. Moreover…

SECTION 4.3 IN POSTED LECTURE NOTES

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…

Determine the cumulative distribution function (cdf) 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

Four Examples: 2 For any b > a > 0, consider the probability density function (pdf)... Determine the cumulative distrib function (cdf) For any it follows that For any it follows that For any it follows that For any it follows that

Four Examples: 2 For any b > a > 0, consider the probability density function (pdf)... Determine the mean Determine the cumulative distrib function (cdf) Determine the variance

WARNING: “IMPROPER INTEGRAL” Four Examples: 3 Consider the following probability density function (pdf)... Confirm pdf WARNING: “IMPROPER INTEGRAL”  1

WARNING: “IMPROPER INTEGRAL” Four Examples: 3 Consider the following probability density function (pdf)... Confirm pdf WARNING: “IMPROPER INTEGRAL” 

WARNING: “IMPROPER INTEGRAL” Four Examples: 3 Consider the following probability density function (pdf)... Confirm pdf WARNING: “IMPROPER INTEGRAL” 

WARNING: “IMPROPER INTEGRAL” Four Examples: 4 Four Examples: 3 Consider the following probability density function (pdf)... Confirm pdf WARNING: “IMPROPER INTEGRAL” 

WARNING: “IMPROPER INTEGRAL” Four Examples: 4 Four Examples: 3 Consider the following probability density function (pdf)... Confirm pdf WARNING: “IMPROPER INTEGRAL”   does not exist!