Continuous Probability Distributions

Statistics for Management and Economics

Chapter 8

Probability Density Functions

Unlike a discrete random variable which we studied in Chapter 7, a continuous random variable is one that can assume an uncountable number of values.

We cannot list the possible values because there is an infinite number of them.

Because there is an infinite number of values, the probability of each individual value is virtually 0.

Point Probabilities are Zero Because there is an infinite number of values,

the probability of each individual value is virtually 0.

Thus, we can determine the probability of a range of values only.

E.g. with a discrete random variable like tossing a die, it is meaningful to talk about P(X=5), say.

In a continuous setting (e.g. with time as a random variable), the probability the random variable of interest, say task length, takes exactly 5 minutes is infinitesimally small, hence P(X=5) = 0. It is meaningful to talk about P(X ≤ 5).

Probability Density FunctionA function f(x) is called a probability density

function (over the range a ≤ x ≤ b if it meets the following requirements:

1) f(x) ≥ 0 for all x between a and b, and

2) The total area under the curve between a and b is 1.0

f(x)

xba

area=1

Characteristics of Density Functions

All density functions must satisfy the following two requirements:

1. The curve must never fall below the horizontal axis. That is,

f(x) ≥ 0 for all x

2. The total area between the curve and the horizontal axis must be 1. In calculus this is expressed as:

f(x) dx = 1

Uniform Distribution Consider the uniform probability

distribution (sometimes called the rectangular probability distribution).

It is described by the function:

f(x)

xba

area = width x height = (b – a) x = 1

*f(x) = 0 for all other values

The Normal Distribution The normal distribution is the most important of

all probability distributions. The probability density function of a normal random variable is given by:

It looks like this: Bell shaped, Symmetrical around the

mean

The Normal DistributionThe normal distribution is fully defined by two parameters:

its standard deviation and mean

Unlike the range of the uniform distribution (a ≤ x ≤ b)Normal distributions range from minus infinity to plus infinity

The normal distribution is bell shaped andsymmetrical about the mean

Standard Normal Distribution A normal distribution whose mean is zero and

standard deviation is one is called the standard normal distribution.

As we shall see shortly, any normal distribution can be converted to a standard normal distribution with simple algebra. This makes calculations much easier.

0

1

1

Normal DistributionThe normal distribution is described by two

parameters:

Its mean and its standard deviation . Increasing the mean shifts the curve to the right…

Normal DistributionThe normal distribution is described by two parameters:

its mean and its standard deviation . Increasing the standard deviation “flattens” the curve…

Calculating Normal ProbabilitiesWe can use the following function

to convert any normal random variable to a standard normal random variable…

Some advice: always draw a

picture!

0

Calculating Normal ProbabilitiesWe can use the following function to

convert any normal random variable to a standard normal random variable…

This shifts the mean of X to

zero…

0

Calculating Normal ProbabilitiesWe can use the following function to

convert any normal random variable to a standard normal random variable…

This changes the shape of the

curve…

0

Calculating Normal Probabilities

We can use Table 3 in

Appendix B to look-up

probabilities P(0 < Z < z)

We can break up P(–.5 < Z < 1) into:

P(–.5 < Z < 0) + P(0 < Z < 1)

The distribution is symmetric around zero, so (multiplying through by minus one and re-arranging the terms) we have:

P(–.5 < Z < 0) = P(.5 > Z > 0) = P(0 < Z < .5)

Hence: P(–.5 < Z < 1) = P(0 < Z < .5) + P(0 < Z < 1)

Calculating Normal ProbabilitiesHow to use Table 3…

This table gives probabilities P(0 < Z < z)

First column = integer + first decimal

Top row = second decimal place

P(0 < Z < 0.5)

P(0 < Z < 1)

P(–.5 < Z < 1) = .1915 + .3414 = .5328

Calculating Normal Probabilities

Recap: The time required to build a computer is normally distributed with a mean of 50 minutes and a standard deviation of 10 minutes

What is the probability that a computer is assembled in a time between 45 and 60 minutes?

P(45 < X < 60) = P(–.5 < Z < 1) = .5328

“Just over half the time, 53% or so, a computer will have an assembly time between 45 minutes and 1

hour”

Using the Normal TableWhat is P(Z > 1.6) ?

0 1.6

P(0 < Z < 1.6) = .4452

P(Z > 1.6) = .5 – P(0 < Z < 1.6)= .5 – .4452

= .0548

z

Using the Normal TableWhat is P(Z < -2.23) ?

0 2.23

P(0 < Z < 2.23)

P(Z < -2.23) = P(Z > 2.23)= .5 – P(0 < Z < 2.23)

= .0129

z

-2.23

P(Z > 2.23)P(Z < -2.23)

Using the Normal TableWhat is P(Z < 1.52) ?

0 1.52

P(Z < 0) = .5

P(Z < 1.52) = .5 + P(0 < Z < 1.52)= .5 + .4357

= .9357

z

P(0 < Z < 1.52)

Using the Normal TableWhat is P(0.9 < Z < 1.9) ?

0 0.9

P(0 < Z < 0.9)

P(0.9 < Z < 1.9) = P(0 < Z < 1.9) – P(0 < Z < 0.9)=.4713 – .3159

= .1554

z

1.9

P(0.9 < Z < 1.9)

Finding Values of Z

Often we’re asked to find some value of Z for a given probability, i.e. given an area (A) under the curve, what is the corresponding value of z (zA) on the horizontal axis that gives us this area? That is:

P(Z > zA) = A

Finding Values of ZWhat value of z corresponds to an area under the

curve of 2.5%? That is, what is z.025 ?

Area = .50 Area = .025

Area = .50–.025 = .4750

If you do a “reverse look-up” on Table 3 for .4750,you will get the corresponding zA = 1.96Since P(z > 1.96) = .025, we say: z.025 = 1.96

Finding and Using Values of Z Excel can be used to find values for normally distributed variables. Syntax: NORMDIST(x,mean,standard_dev,cumulative)

Returns the normal distribution for the specified mean and standard deviation.

X is the value for which you want the distribution. Mean is the arithmetic mean of the distribution. Standard_dev is the standard deviation of the distribution. Cumulative is a logical value that determines the form of the

function. If cumulative is TRUE, NORMDIST returns the cumulative distribution function; if FALSE, it returns the probability mass function.

Syntax: NORMSDIST(z) Z is the value for which you want the distribution. Returns the standard normal cumulative distribution function.

The distribution has a mean of 0 (zero) and a standard deviation of one. Use this function in place of a table of standard normal curve areas.

Finding and Using Values of Z

Other Z values are Z.05 = 1.645 Z.01 = 2.33

Because z.025 = 1.96 and - z.025= -1.96, it follows that we can stateP(-1.96 < Z < 1.96) = .95

Similarly, P(-1.645 < Z < 1.645) = .90

Exponential Distribution Another important continuous distribution is

the exponential distribution Closely related to the discrete Poisson

distribution Describes a probability distribution of the

times between random occurrences Skewed to the right The x values range from zero to infinity Curve steadily decreases as x gets larger The apex (top of the curve) is always at x=0

Exponential Distribution The exponential distribution has this probability

density function:

Note that x ≥ 0. Time (for example) is a non-negative quantity; the exponential distribution is often used for time related phenomena such as the length of time between phone calls or between parts arriving at an assembly station.

Note also that the mean and standard deviation are equal to each other and to the inverse of the parameter of the distribution (lambda )

Exponential Distribution

The exponential distribution depends upon the value of

Smaller values of “flatten” the curve:

(E.g. exponential distributions for = .5, 1, 2)

Exponential DistributionRecall from calculus that, in order to find the area under the curve,

we take the integral of the probability density function over all real values of x, f(x) = e- x . If X is an exponential random variable, then we can calculate probabilities by:

f(x) = e - x

P(x ≥ x0) = e - x

And…

Exponential Distribution Excel can be used to find probabilities for the

exponential distribution Syntax: EXPONDIST(x,lambda,cumulative)

X is the value of the function. Lambda is the parameter value. Cumulative is a logical value that indicates which

form of the exponential function to provide. If cumulative is TRUE, EXPONDIST returns the cumulative distribution function; if FALSE, it returns the probability density function.

Returns the exponential distribution.

Other Continuous Distributions

There are three other important continuous distributions which will be used extensively in later sections:

Student’s t Distribution Chi-Squared Distribution F Distribution

Student t Distribution

Much like the standard normal distribution, the Student t distribution is bell-shaped and symmetrical about its mean of zero.

In much the same way that and define the normal distribution, , the degrees of freedom, defines the Student t Distribution:

As the number of degrees of freedom increases, the t distribution approaches the standard normal distribution.

2 Chi-Squared Distribution Positively skewed continuous

distribution As before, based on , the number of

degrees of freedom.

F Distribution

F > 0. Two parameters define this distribution, and like we’ve already seen these are again degrees of freedom (df).

1 is the “numerator” degrees of freedom and 2 is the “denominator” degrees of freedom.

The F distribution is similar to the 2 distribution in that its starts at zero (is non-negative) and is not symmetrical.