Chapter 4 - Lecture 1 Probability Density Functions and Cumul. Distribution Functions

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Chapter 4 - Lecture 1

Probability Density Functions and Cumulative Distribution Functions

Andreas Artemiou

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Continuous random variables Review

Probability distribution function Example

Uniform Distribution Definition Example

Cumulative distribution function Definition

Example Useful results

Relationship between the pdf and the cdf

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Review

Probability distribution function Example

Review

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In Chapter 3 we have seen that a continuous random variable is one that can take any possible value in a given interval.

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Examples

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People weight

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People height

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Distance between two cities

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Review

Probability distribution function Example

Probability distribution functon

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Now if X is continuous random variable the probability distribution or probability density function (pdf) of X is a function f (x ) such that

P(a ≤ X ≤ b) = Z

a

f (x )dx

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Review

Probability distribution function Example

Legitimate pdf

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A function is a legitimate pdf if it satisfies the following two conditions

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f (x ) ≥ 0∀X

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Z

−∞

f (x )dx = 1

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Review

Probability distribution function Example

Important property

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Note that in the continuous case P(X = c) = 0 for every possible value of c. (why?)

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This has a very useful consequence in the continuous case:

P(a ≤ X ≤ b) = P(a < X ≤ b) = P(a ≤ X < b) = P(a < X < b)

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Review

Probability distribution function Example

Example 4.5 page 158

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Definition Example

Uniform Distribution

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A continuous random variable X is said to have a uniform distribution on the interval [A, B] if the pdf of X is the following:

f (x ) =





 1

B − A , A ≤ X ≤ B

0 , otherwise

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Definition Example

Example

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If in a Friday quiz we denote with X the time that the first student will finish and X follows a uniform distribution in the interval 5 to 15 minutes.

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Find the probability distribution

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Find the P(X < 4), P(X < 12) and P(X > 7).

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Definition Example Useful results

Relationship between the pdf and the cdf

Cumulative distribution function

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The cumulative distribution function (cdf) for a continuous random variable X is the following:

F (x ) = P(X ≤ x ) = Z

−∞

f (y )dy

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Definition Example Useful results

Relationship between the pdf and the cdf

Example

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If in a Friday quiz we denote with X the time that the first student will finish and X follows a uniform distribution in the interval 5 to 15 minutes.

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Find the cumulative distribution function of X .

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Definition Example Useful results

Relationship between the pdf and the cdf

Example

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In the continuous case is very useful to use the cdf to find probabilities using the formulas:

P(X > a) = 1 − F (a)

P(a ≤ X ≤ b) = F (b) − F (a)

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Definition Example Useful results

Relationship between the pdf and the cdf

Example

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If in a Friday quiz we denote with X the time that the first student will finish and X follows a uniform distribution in the interval 5 to 15 minutes.

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Find P(X > 7) and P(6 < X < 11).

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Definition Example Useful results

Relationship between the pdf and the cdf

Obtaining f (x ) from F (x )

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If X is a continuous random variable with pdf f (x ) and cdf F (x ), then at every x at which the derivative of F (x ), denoted with F

(x ), exists we have that F

(x ) = f (x ).

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Prove this for the quiz example in the previous slide.

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Definition Example

Percentiles

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If p is a number between 0 and 1. Then the (100p)th

percentile of the distribution of a continuous random variable X is denoted by η(p) and it satisfies the following:

p = F ((η(p))) = Z

−∞

f (y )dy

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Definition Example

Example 4.9 page 164

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Definition Example

Median

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By definition the median is the middle observation. When we

have a continuous random variable the median is the same as

the 50th percentile. So half the area under the curve is below

the median (on the left) and half above the median (on the

right).

Outline Continuous random variables Uniform Distribution Cumulative distribution function Percentiles Exercises

Exercises

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Section 4.1 page 165

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Exercises 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17