Continuous Random Variables Lecture 22 Section Mon, Feb 25, 2008

Random Variables Random variable Discrete random variable Continuous random variable

Continuous Probability Distribution Functions Continuous Probability Distribution Function (pdf) – For a random variable X, it is a function with the property that the area below the graph of the function between any two points a and b equals the probability that a ≤ X ≤ b. Remember, AREA = PROPORTION = PROBABILITY

Example The TI-83 will return a random number between 0 and 1 if we enter rand and press ENTER. These numbers have a uniform distribution from 0 to 1. Let X be the random number returned by the TI-83.

Example The graph of the pdf of X. x f(x)f(x) 01 1

Example What is the probability that the random number is at least 0.3?

Example What is the probability that the random number is at least 0.3? x f(x)f(x)

Example What is the probability that the random number is at least 0.3? x f(x)f(x)

Example What is the probability that the random number is at least 0.3? x f(x)f(x)

Area = 0.7 Example Probability = 70%. x f(x)f(x)

0.25 Example What is the probability that the random number is between 0.25 and 0.75? x f(x)f(x)

Example What is the probability that the random number is between 0.25 and 0.75? x f(x)f(x)

Example What is the probability that the random number is between 0.25 and 0.75? x f(x)f(x)

Example Probability = 50%. x f(x)f(x) Area = 0.5

Uniform Distributions The uniform distribution from a to b is denoted U(a, b). ab 1/(b – a) x

A Non-Uniform Distribution Consider this distribution. 510 x

A Non-Uniform Distribution What is the height? 510 ? x

A Non-Uniform Distribution The height is x

A Non-Uniform Distribution What is the probability that 6  X  8? x 68

A Non-Uniform Distribution It is the same as the area between 6 and x 68

Uniform Distributions The uniform distribution from a to b is denoted U(a, b). ab 1/(b – a)

Hypothesis Testing (n = 1) An experiment is designed to determine whether a random variable X has the distribution U(0, 1) or U(0.5, 1.5).  H 0 : X is U(0, 1).  H 1 : X is U(0.5, 1.5). One value of X is sampled (n = 1).

Hypothesis Testing (n = 1) An experiment is designed to determine whether a random variable X has the distribution U(0, 1) or U(0.5, 1.5).  H 0 : X is U(0, 1).  H 1 : X is U(0.5, 1.5). One value of X is sampled (n = 1). If X is more than 0.75, then H 0 will be rejected.

Hypothesis Testing (n = 1) Distribution of X under H 0 : Distribution of X under H 1 :

Hypothesis Testing (n = 1) What are  and  ?

Hypothesis Testing (n = 1) What are  and  ?

Hypothesis Testing (n = 1) What are  and  ? Acceptance RegionRejection Region

Hypothesis Testing (n = 1) What are  and  ?

Hypothesis Testing (n = 1) What are  and  ?   = ¼ = 0.25

Hypothesis Testing (n = 1) What are  and  ?   = ¼ = 0.25   = ¼ =

Example Now suppose we use the TI-83 to get two random numbers from 0 to 1, and then add them together. Let X 2 = the average of the two random numbers. What is the pdf of X 2 ?

Example The graph of the pdf of X 2. y f(y)f(y) ?

Example The graph of the pdf of X 2. y f(y)f(y) Area = 1

Example What is the probability that X 2 is between 0.25 and 0.75? y f(y)f(y)

Example What is the probability that X 2 is between 0.25 and 0.75? y f(y)f(y)

Example The probability equals the area under the graph from 0.25 to y f(y)f(y)

Example Cut it into two simple shapes, with areas 0.25 and 0.5. y f(y)f(y) Area = 0.5 Area =

Example The total area is y f(y)f(y) Area = 0.75

Hypothesis Testing (n = 2) An experiment is designed to determine whether a random variable X has the distribution U(0, 1) or U(0.5, 1.5).  H 0 : X is U(0, 1).  H 1 : X is U(0.5, 1.5). Two values of X are sampled (n = 2). Let X 2 be the average. If X 2 is more than 0.75, then H 0 will be rejected.

Hypothesis Testing (n = 2) Distribution of X 2 under H 0 : Distribution of X 2 under H 1 :

Hypothesis Testing (n = 2) What are  and  ?

Hypothesis Testing (n = 2) What are  and  ?

Hypothesis Testing (n = 2) What are  and  ?

Hypothesis Testing (n = 2) What are  and  ?   = 1/8 = 0.125

Hypothesis Testing (n = 2) What are  and  ?   = 1/8 =   = 1/8 = 0.125

Conclusion By increasing the sample size, we can lower both  and  simultaneously.