Continuous Random Variables

Lecture 22

Section 7.5.4

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)

0 1

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)

0 1

1

0.3

Example

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

x

f(x)

0 1

1

0.3

Example

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

x

f(x)

0 1

1

0.3

Area = 0.7

Example

Probability = 70%.

x

f(x)

0 1

1

0.3

0.25

Example

What is the probability that the random number is between 0.25 and 0.75?

x

f(x)

0 1

1

0.75

Example

What is the probability that the random number is between 0.25 and 0.75?

x

f(x)

0 1

1

0.25 0.75

Example

What is the probability that the random number is between 0.25 and 0.75?

x

f(x)

0 1

1

0.25 0.75

Example

Probability = 50%.

x

f(x)

0 1

1

0.25 0.75

Area = 0.5

Uniform Distributions

The uniform distribution from a to b is denoted U(a, b).

a b

1/(b – a)

x

A Non-Uniform Distribution

Consider this distribution.

5 10x

A Non-Uniform Distribution

What is the height?

5 10

?

x

A Non-Uniform Distribution

The height is 0.4.

5 10

0.4

x

A Non-Uniform Distribution

What is the probability that 6 X 8?

5 10

0.4

x6 8

A Non-Uniform Distribution

It is the same as the area between 6 and 8.

5 10

0.4

x6 8

Uniform Distributions

The uniform distribution from a to b is denoted U(a, b).

a b

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).H0: X is U(0, 1).

H1: 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).H0: X is U(0, 1).

H1: X is U(0.5, 1.5).

One value of X is sampled (n = 1). If X is more than 0.75, then H0 will be

rejected.

Hypothesis Testing (n = 1)

Distribution of X under H0:

Distribution of X under H1:

0 0.5 1 1.5

1

0 0.5 1 1.5

1

Hypothesis Testing (n = 1)

What are and ?

0 0.5 1 1.5

1

0 0.5 1 1.5

1

Hypothesis Testing (n = 1)

What are and ?

0.75

0.75

0 0.5 1 1.5

1

0 0.5 1 1.5

1

Hypothesis Testing (n = 1)

What are and ?

0.75

0.75

0 0.5 1 1.5

1

0 0.5 1 1.5

1

Acceptance Region Rejection Region

Hypothesis Testing (n = 1)

What are and ?

0.75

0.75

0 0.5 1 1.5

1

0 0.5 1 1.5

1

Hypothesis Testing (n = 1)

What are and ?

0.75

0.75

0 0.5 1 1.5

1

0 0.5 1 1.5

1

= ¼ = 0.25

Hypothesis Testing (n = 1)

What are and ?

0.75

0 0.5 1 1.5

1

0 0.5 1 1.5

1

= ¼ = 0.25

= ¼ = 0.25

0.75

Example

Now suppose we use the TI-83 to get two random numbers from 0 to 1, and then add them together.

Let X2 = the average of the two random numbers.

What is the pdf of X2?

Example

The graph of the pdf of X2.

y

f(y)

0 0.5 1

?

Example

The graph of the pdf of X2.

y

f(y)

0 0.5 1

2

Area = 1

Example

What is the probability that X2 is between 0.25 and 0.75?

y

f(y)

0 0.5 10.25 0.75

2

Example

What is the probability that X2 is between 0.25 and 0.75?

y

f(y)

0 0.5 10.25 0.75

2

Example

The probability equals the area under the graph from 0.25 to 0.75.

y

f(y)

0 0.5

2

10.25 0.75

Example

Cut it into two simple shapes, with areas 0.25 and 0.5.

y

f(y)

0 0.5 10.25 0.75

2

Area = 0.5

Area = 0.250.5

Example

The total area is 0.75.

y

f(y)

0 0.5 10.25 0.75

2

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). H0: X is U(0, 1).

H1: X is U(0.5, 1.5).

Two values of X are sampled (n = 2). Let X2 be the average.

If X2 is more than 0.75, then H0 will be rejected.

Hypothesis Testing (n = 2)

Distribution of X2 under H0:

Distribution of X2 under H1:0 0.5 1 1.5

2

0 0.5 1 1.5

2

Hypothesis Testing (n = 2)

What are and ?

0 0.5 1 1.5

2

0 0.5 1 1.5

2

Hypothesis Testing (n = 2)

What are and ?

0 0.5 1 1.5

2

0 0.5 1 1.5

2

0.75

0.75

Hypothesis Testing (n = 2)

What are and ?

0 0.5 1 1.5

2

0 0.5 1 1.5

2

0.75

0.75

Hypothesis Testing (n = 2)

What are and ?

0 0.5 1 1.5

2

0 0.5 1 1.5

2

0.75

0.75

= 1/8 = 0.125

Hypothesis Testing (n = 2)

What are and ?

0 0.5 1 1.5

2

0 0.5 1 1.5

2

0.75

0.75

= 1/8 = 0.125

= 1/8 = 0.125

Conclusion

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