Chapter 4: Continuous Random Variables and Probability Distributions

4-1 Continuous Random Variables4-2 Probability Distributions and Probability Density Functions4-3 Cumulative Distribution Functions4-4 Mean and Variance of a Continuous Random Variable4-5 Continuous Uniform Distribution 4-6 Normal Distribution4-7 Normal Approximation to the Binomial and Poisson

Distributions4-8 Exponential Distribution4-9 Erlang and Gamma Distributions4-10 Weibull Distribution4-11 Lognormal Distribution4-12 Beta Distribution 1

Chapter Learning ObjectivesAfter careful study of this chapter you should be able to:1.Determine probabilities from probability density functions 2.Determine probabilities from cumulative distribution functions and cumulative distribution functions from probability density functions, and the reverse 3.Calculate means and variances for continuous random variables 4.Understand the assumptions for some common continuous probability distributions 5.Select an appropriate continuous probability distribution to calculate probabilities in specific applications 6.Calculate probabilities, determine means and variances for some common continuous probability distributions 7.Standardize normal random variables 8.Use the table for the cumulative distribution function of a standard normal distribution to calculate probabilities 9.Approximate probabilities for some binomial and Poisson distributions

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Continuous Random Variables

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Probability Distributions and Probability Density Functions

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Probability Density Function Defined

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Probability Density Functions and Histograms

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A Probability Density Function Example

Let the continuous random variable X denote the current measured in a thin copper wire in milliamperes. Assume that the range of X is [0, 20 mA], and assume that the probability density function of X is f(x) = 0.05 for 0 � x � 20. What is the probability that a current measurement is less than 10 milliamperes?

The probability density function is shown in the figure below (it is assumed that f(x) = 0 wherever it is not specifically defined). The probability requested is indicated by the shaded area in the figure, which can be computed using the equation from slide #5..

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Example 4-1

Another Probability Density Function Example

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Example 4-2

The Cumulative Distribution Function in the Continuous Case

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A Cumulative Distribution Function Example

Example 4-4

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Cumulative Distribution Function Examples

• Here is the cdf for Example 4-2:

• Here is the cdf for Example 4-1:

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Mean and Variance of a Continuous Random Variable

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Example of Mean and Variance of a Continuous Random Variable

Example 4-6

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The Mean of a Function of a Continuous Random Variable

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The Continuous Uniform Distribution

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The Mean and Variance of the Continuous Uniform Distribution

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• Use these formula for the situation in example 4-1 (and 4-6) to verify the results from slide #13 that when a=0 and b=20:

� = 10.0 and �2 = 33.33

The cdf of the General Continuous Uniform Distribution

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The Normal Distribution

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Example of Normal Distribution pdfs

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• For any normal random variable:

Well-Known Normal Distribution Probabilities

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The Standard Normal Distribution

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Example 4-11

Figure 4-13 Standard normal probability density function.

How to Use a Table of the cdf of the Standard Normal Distribution

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Cumulative Standard Normal Distribution

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

1.0 0.841345 0.843752 0.846136 0.848495 0.850830 0.853141 0.855428 0.857690 0.859929 0.8621431.1 0.864334 0.866500 0.868643 0.870762 0.872857 0.874928 0.876976 0.878999 0.881000 0.8829771.2 0.884930 0.886860 0.888767 0.890651 0.892512 0.894350 0.896165 0.897958 0.899727 0.9014751.3 0.903199 0.904902 0.906582 0.908241 0.909877 0.911492 0.913085 0.914657 0.916207 0.9177361.4 0.919243 0.920730 0.922196 0.923641 0.925066 0.926471 0.927855 0.929219 0.930563 0.9318881.5 0.933193 0.934478 0.935744 0.936992 0.938220 0.939429 0.940620 0.941792 0.942947 0.9440831.6 0.945201 0.946301 0.947384 0.948449 0.949497 0.950529 0.951543 0.952540 0.953521 0.9544861.7 0.955435 0.956367 0.957284 0.958185 0.959071 0.959941 0.960796 0.961636 0.962462 0.9632731.8 0.964070 0.964852 0.965621 0.966375 0.967116 0.967843 0.968557 0.969258 0.969946 0.9706211.9 0.971283 0.971933 0.972571 0.973197 0.973810 0.974412 0.975002 0.975581 0.976148 0.976705

Example 4-12: Standard Normal Exercises1. P(Z > 1.26) = 0.1038

2. P(Z < -0.86) = 0.195

3. P(Z > -1.37) = 0.915

4. P(-1.25 < 0.37) =

0.5387

5. P(Z � -4.6) � 0

6. Find z for P(Z � z) =

0.05, z = -1.65

7. Find z for (-z < Z < z)

= 0.99, z = 2.58

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Figure 4-14 Graphical displays for standard normal distributions.

Using The Standard Normal cdf to Determine Probabilities for Other Normal Distributions

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Example 4-13

An Example on Standardizing a Normal Random Variable

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Another Example (4-14) on Standardizing A Normal Random Variable

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The Normal Approximation to the Binomial Distribution

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150+.05-160

-0.75) 0.75) 0.773

An Example (4-17 & 4-18) of a Normal Approximation to the Binomial

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Guidelines for Using the Normal to Approximate the Binomial or

Hypergeometric

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The Normal Approximation to the Poisson Distribution

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An Example of a Normal Approximation to the Poisson

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Example 4-20

The Exponential Distribution

The exponential distribution cdf is:F(x) = 1-e-�x

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An Exponential Distribution Example (Example 4-21)

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An Exponential Distribution Example (Example 4-21 continued)

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An Exponential Distribution Example (Example 4-21 continued)

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e.g.In Example 4-21, suppose that there are no log-ons from 12:00 to 12:15; the probability that there are no log-ons from 12:15 to 12:21 is still 0.082. Because we have already been waiting for 15 minutes, we feel that we are “due.” That is, the probability of a log-on in the next 6 minutes should be greater than 0.082. However, for an exponential distribution this is not true.

An Unusual Property of the Exponential Distribution

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Lack of Memory Property

Exponential Application in Reliability

• The reliability of electronic components is often modeled by the exponential distribution. A chip might have mean time to failure of 40,000 operating hours

• The memoryless property implies that the component does not wear out – the probability of failure in the next hour is constant, regardless of the component age

• The reliability of mechanical components do have a memory – the probability of failure in the next hour increases as the component ages. The Weibull distribution is used to model this situation

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The Erlang Distribution

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The random variable X that equals the interval length until rcounts occur in a Poisson process with mean � > 0 has and Erlang random variable with parameters � and r. The probability density function of X is

for x > 0 and r =1, 2, 3, ….

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Example pdfs for the Erlang Distribution

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

Misc. Erlang pdfs for different values of r (the "order") {�=0.5}

r=1 r=2 r=4 r=8

Correction!2

The Gamma Distribution

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Figure 4-25 Gammaprobability density functions for selected values of r and �.

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Example pdfs for the Gamma Distribution

The Weibull Distribution

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Figure 4-26 Weibullprobability density functions for selected values of � and �.

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Example pdfs for the Weibull Distribution

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The Lognormal Distribution

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Example pdfs for the Lognormal Distribution

• A continuous distribution that is flexible, but bounded over the [0, 1] interval is useful for probability models. Examples are:– Proportion of solar radiation absorbed by a material– Proportion of the max time to complete a task

Beta Distribution

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� � � �� � � � � � 11

The random variable with probability density function

= 1 for 0 1

is a beta random variable with parameters 0 and 0.

X

f x x x x��� �� �

� �

��� �

� ��

� �

� �

� �� � � �

22 1

E X

V X

� � �

���� � � �

� �

� �

If X has a beta distribution with parameters � and �, the mean and variance of X are:

Beta Shapes are Flexible

• Distribution shape guidelines:– If � = �, symmetrical

about x = 0.5– If � = � = 1, uniform– If � = � < 1, symmetric

& U- shaped– If � = � > 1, symmetric

& mound-shaped– If � � �, skewed

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Figure 4-28 Beta probability density functions for selected values of the parameters � and �

Extended Range for the Beta Distribution

The beta random variable X is defined for the [0, 1] interval. That interval can be changed to [a, b]. Then the random variable W is defined as a linear function of X:

W = a + (b –a)XWith mean and variance:

E(W) = a + (b –a) E(X)V(W) = (b - a)2V(X)

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