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Continuous Probability Distributions

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Continuous Probability Distributions

Statistics for Management and Economics

Chapter 8

- 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.

- 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).

f(x)

area=1

a

b

x

A function f(x) is called a probability density function (over the range a ≤ x ≤ b if it meets the following requirements:

- f(x) ≥ 0 for all x between a and b, and
- The total area under the curve between a and b is 1.0

All density functions must satisfy the following two requirements:

- The curve must never fall below the horizontal axis. That is,f(x) ≥ 0 for all x
- The total area between the curve and the horizontal axis must be 1. In calculus this is expressed as: f(x) dx = 1

f(x)

a

b

x

area = width x height = (b – a) x = 1

- Consider the uniform probability distribution (sometimes called the rectangular probability distribution).
- It is described by the function:

*f(x) = 0 for all other values

- 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 distribution is fully defined by two parameters:

its standard deviation andmean

The normal distribution is bell shaped and

symmetrical about the mean

Unlike the range of the uniform distribution (a ≤ x ≤ b)

Normal distributions range from minus infinity to plus infinity

0

1

1

- 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.

The normal distribution is described by two parameters:

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

The normal distribution is described by two parameters:

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

We can use the following function to convert any normal random variable to a standard normal random variable…

0

Some advice: always draw a picture!

We can use the following function to convert any normal random variable to a standard normal random variable…

0

This shifts the mean of X to zero…

We can use the following function to convert any normal random variable to a standard normal random variable…

0

This changes the shape of the curve…

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)

How 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

- 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”

What is P(Z > 1.6) ?

P(0 < Z < 1.6) = .4452

z

0

1.6

P(Z > 1.6) = .5 – P(0 < Z < 1.6)

= .5 – .4452

= .0548

P(0 < Z < 2.23)

What is P(Z < -2.23) ?

P(Z < -2.23)

P(Z > 2.23)

z

0

-2.23

2.23

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

= .5 – P(0 < Z < 2.23)

= .0129

What is P(Z < 1.52) ?

P(0 < Z < 1.52)

P(Z < 0) = .5

z

0

1.52

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

= .5 + .4357

= .9357

P(0 < Z < 0.9)

What is P(0.9 < Z < 1.9) ?

P(0.9 < Z < 1.9)

z

0

0.9

1.9

P(0.9 < Z < 1.9) = P(0 < Z < 1.9) – P(0 < Z < 0.9)

=.4713 – .3159

= .1554

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

Area = .50

Area = .025

Area = .50–.025 = .4750

What value of z corresponds to an area under the curve of 2.5%? That is, what is z.025 ?

If you do a “reverse look-up” on Table 3 for .4750,you will get the corresponding zA = 1.96

Since P(z > 1.96) = .025, we say: z.025 = 1.96

- 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.

- 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 state
P(-1.96 < Z < 1.96) = .95

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

- 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

- 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 )

- The exponential distribution depends upon the value of
- Smaller values of “flatten” the curve:
- (E.g. exponential distributions for = .5, 1, 2)

Recall 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…

- 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.

- There are three other important continuous distributions which will be used extensively in later sections:
- Student’s t Distribution
- Chi-Squared Distribution
- F 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.

- Positively skewed continuous distribution
- As before, based on , the number of degrees of freedom.

- 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 2is 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.