Chapter

Seven

McGraw-Hill/Irwin

© 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.

Continuous Probability Distributions

Continuous Probability Distributions

•Probability is calculated for a range of values and not a specific value.

•Eg. You don’t ask what is the probability drive time taken by students coming to CSUN is (exactly) 7 minutes?

Rather, you ask, what is the probability drive time is between, say, 5 – 10 minutes?

The Random Variable (RV) that can take infinite values within a given range.

If the slots are infinitely narrow, then the chance of winning any one slot is zero (=1/∞)

So, we will calculate probabilities for a range of values, rather than for a value of the RV.We will take the ratio of the area for the range of values to the total area.

The NormalNormal probability distribution

Two characteristics determine a normal curve:

Mean

Std. Deviation

∞ ∞

Family of Normal Distributions

See this in Visual Statistics

Calculating area under a Normal curve

Remember, in calculating probabilities for continuous RVs, we have to measure the area under the curve.

Providing tables of areas for every possible normal curve is impractical (because too many combinations of and ) .

Fortunately, there is one member of the family that can be used to calculate areas for any normal curve.

Its mean is 0 and standard deviation is 1.

This curve is called the Standard Normal Distribution or the Z-Distribution. (Appendix D, Page 496)

Standard Normal Distribution or the Z-Distribution.

Question: How do we transform any given normal curve to a Standard Normal (or Z) distribution ?

Slide Squish

X

z

Remember the Transformation formula !!!

Notice that Z is a measure of how many SDs a given X value is from the mean.

The original normal curve with mean and s.d. now becomes …

… the Standard Normal Distribution with mean 0 and s.d. 1 !

Source file: DemoX-MuBySigma.xls

Excel Example

Problem on Page 197

Weekly income of a foreman is normally distributed with mean of $1000 and s.d. of $100. You select a foreman randomly? What is the probability that he/she earns between $1000-1100?

Use the Z-transformation formula and compute the value of Z corresponding to X=1000 and X=1100.

When X=1000, Z is (1000-1000)/100 = 0When X=1100, Z is (1100-1000)/100 = 1

Slide

μ=0 100

σ=100

μ=1000 1100

σ=100

Z=0 1

σ=1Proportion of area between dotted lines to the total area of the curve does not change with slide & squish operation!

Equivalent Std Normal

Curve

The area (which represents the probability) is 0.3413

Go to Appendix D on Page 496 and find the area under the curve between Z=0 and Z=1.

Variation of the same problem ( Page 198)

What is the probability of randomly selecting a foreman who earned less than $1100?

The answer is: 0.8413

Another variation of the same problem (Page 198-199)

a. What is the probability of selecting a foreman whose income is between $790-1000?

Z = (790-1000)/100 = -2.10From the Appendix table, this gives 0.4821

b. Less than $790?

Z value is same as above. But we compute 0.5 – 0.4821 = 0.0179

What is the area between $840-1200? (page 200)

Corresponding to $840, Z is (840-1000)/100 = -1.60Corresponding to $1200, Z is (1200-1000)/100 = 2.0

Use Appendix D to find the total area. The answer is .4452+.4772 = 0.9224

Calculate the area under the curve between $1150-1250?

Corresponding to $1250, Z is (1250-1000)/100 = 2.50Corresponding to $1150, Z is (1150-1000)/100 = 1.50

Use the Appendix D to find the answer to be 0.0606