Continuous Continuous Random Random VariablesVariables

Lecture 25Lecture 25

Section 7.5.3Section 7.5.3

Mon, Feb 28, 2005Mon, Feb 28, 2005

Continuous Probability Continuous Probability Distribution FunctionsDistribution Functions

Continuous Probability Distribution Continuous Probability Distribution Function (pdf)Function (pdf) – For a random – For a random variable variable XX, it is a function with the , it is a function with the property that the property that the areaarea between the between the graph of the function and an interval graph of the function and an interval aa ≤ ≤ xx ≤ ≤ bb equals the equals the probabilityprobability that that aa ≤ ≤ XX ≤ ≤ bb..

In other words, In other words,

AREA = PROBABILITYAREA = PROBABILITY

ExampleExample

The TI-83 will return a random The TI-83 will return a random number between 0 and 1 if we enter number between 0 and 1 if we enter rand and press ENTER.rand and press ENTER.

These numbers have a uniform These numbers have a uniform distribution from 0 to 1.distribution from 0 to 1.

Let Let XX be the random number be the random number returned by the TI-83.returned by the TI-83.

ExampleExample

The graph of the pdf of The graph of the pdf of XX..

x

f(x)

0 1

1

ExampleExample

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

ExampleExample

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

x

f(x)

0 1

1

0.3

ExampleExample

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

x

f(x)

0 1

1

0.3

ExampleExample

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

x

f(x)

0 1

1

0.3

Area = 0.7

ExampleExample

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

Probability = 70%.Probability = 70%.

x

f(x)

0 1

1

0.3

ExampleExample

What is the probability that the What is the probability that the random number is between 0.3 and random number is between 0.3 and 0.9?0.9?

x

f(x)

0 1

1

0.3 0.9

ExampleExample

What is the probability that the What is the probability that the random number is between 0.3 and random number is between 0.3 and 0.9?0.9?

x

f(x)

0 1

1

0.3 0.9

ExampleExample

What is the probability that the What is the probability that the random number is between 0.3 and random number is between 0.3 and 0.9?0.9?

x

f(x)

0 1

1

0.3 0.9

Area = 0.6

ExampleExample

What is the probability that the What is the probability that the random number is between 0.3 and random number is between 0.3 and 0.9?0.9?

Probability = 60%.Probability = 60%.

x

f(x)

0 1

1

0.3 0.9

ExperimentExperiment

Use the TI-83 to generate 500 values Use the TI-83 to generate 500 values of of XX.. Use rand(500) to do this.Use rand(500) to do this.

Check to see what proportion of Check to see what proportion of them are between 0.3 and 0.9.them are between 0.3 and 0.9. Use a TI-83 histogram and Trace to do Use a TI-83 histogram and Trace to do

this.this.

ExampleExample

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

Let Let YY = the sum of the two random = the sum of the two random numbers.numbers.

What is the pdf of What is the pdf of YY??

ExampleExample

The graph of the pdf of The graph of the pdf of YY..

y

f(y)

0 1 2

1

ExampleExample

The graph of the pdf of Y.The graph of the pdf of Y.

y

f(y)

0 1 2

1

Area = 1

ExampleExample

What is the probability that What is the probability that YY is is between 0.5 and 1.5?between 0.5 and 1.5?

y

f(y)

0 1 20.5 1.5

1

ExampleExample

What is the probability that What is the probability that YY is is between 0.5 and 1.5?between 0.5 and 1.5?

y

f(y)

0 1 20.5 1.5

1

ExampleExample

The probability equals the area The probability equals the area under the graph from 0.5 to 1.5.under the graph from 0.5 to 1.5.

y

f(y)

0 1

1

20.5 1.5

ExampleExample

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

y

f(y)

0 1 20.5 1.5

1

Area = 0.5

Area = 0.250.5

ExampleExample

The total area is 0.75.The total area is 0.75. The probability is 75%.The probability is 75%.

y

f(y)

0 1 20.5 1.5

1

Area = 0.75

VerificationVerification

Use the TI-83 to generate 500 values Use the TI-83 to generate 500 values of of YY.. Use rand(500) + rand(500).Use rand(500) + rand(500).

Use a histogram to find out how Use a histogram to find out how many are between 0.5 and 1.5.many are between 0.5 and 1.5.

ExampleExample

Suppose we get 12 random Suppose we get 12 random numbers, uniformly distributed numbers, uniformly distributed between 0 and 1, from the TI-83 and between 0 and 1, from the TI-83 and add them all up.add them all up.

Let Let XX = sum of 12 random numbers = sum of 12 random numbers from 0 to 1.from 0 to 1.

What is the pdf of What is the pdf of XX??

ExampleExample

It turns out that the pdf of It turns out that the pdf of XX is is approximately normal with a mean approximately normal with a mean of 6 and a standard of 1.of 6 and a standard of 1.

x6 7 8 9543

N(6, 1)

ExampleExample

What is the probability that the sum What is the probability that the sum will be between 5 and 7?will be between 5 and 7?

P(5 < P(5 < XX < 7) = P(–1 < < 7) = P(–1 < ZZ < 1) < 1)

= 0.8413 – 0.1587= 0.8413 – 0.1587

= 0.6826.= 0.6826.

ExampleExample

What is the probability that the sum What is the probability that the sum will be between 4 and 8?will be between 4 and 8?

P(4 < P(4 < XX < 8) = P(–2 < < 8) = P(–2 < ZZ < 2) < 2)

= 0.9772 – 0.0228= 0.9772 – 0.0228

= 0.9544.= 0.9544.

ExperimentExperiment

Use the Excel spreadsheet Use the Excel spreadsheet Uniform12.xlsUniform12.xls to generate 1000 to generate 1000 values of values of XX, where , where XX is the sum of 12 is the sum of 12 random numbers from random numbers from UU(0, 1).(0, 1). We should see a value between 5 and 7 We should see a value between 5 and 7

about 68% of the time.about 68% of the time. We should see a value between 4 and 8 We should see a value between 4 and 8

about 95% of the time.about 95% of the time. We should see a value between 3 and 9 We should see a value between 3 and 9

nearly always (99.7%).nearly always (99.7%).