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Chapter 8 – Further Applications of Integration

8.5 Probability

8.5 Probability Erickson

8.5 Probability2

Definitions Let’s consider the cholesterol level of a person chosen at

random from a certain age group or the height of an adult male or female chosen at random. These quantities are called continuous random variable because their values actually range over an interval of real numbers even though they might be recorded only to the nearest integer.

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Definitions Every continuous random variable X has a probability density

function f. This means that the probability that X lies between a and b is found by integrating f from a to b.

Because probabilities are measured on a scale from 0 to 1, it follows that

( ) ( )b

a

P a X b f x dx

( ) 1b

a

f x dx

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Example 1 – pg. 573 #4 Let if x 0 and f (x) = 0 if x < 0.

Verify that f is a probability density function.

Find P(1 X 2).

( ) xf x xe

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Average Values The mean of any probability density function f is defined

to be

This mean can be interpreted as the long-run average value of the random variable X. It can also be interpreted as a measure of centrality of the probability density function.

( )x f x dx

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Mean If is the region that lies under the graph of f, we know

from section 8.3 that the x-coordinate of the centroid of is

So a thin plate in the shape of balances at a point on the vertical line x = .

( )

( )

( )

x f x dx

x x f x dx

f x dx

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Median Another measure of a central probability density function

is the median. In general, the median of a probability density function is the number m such that

1( )

2m

f x dx

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Example 2 – pg. 574 #9 Suppose the average waiting time for a customer’s call to

be answered by a company representative is five minutes. Show that the median waiting time for a phone company is about 3.5 minutes.

51( )

5

t

f t e

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Normal Distribution The normal distribution is a continuous probability

distribution that often gives a good description of data that cluster around the mean. The probability density function of the random variable X is a member of the family of functions

The positive constant is the standard deviation. It measures how spread out the values of X are.

2

221( )

2

x

f x e

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Normal Distribution We can see how the graph changes as changes.

We can say that

2

2211

2

x

e

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Example 3 – pg. 574 # 10 A type of light bulb is labeled as having an

average lifetime of 1000 hours. It’s reasonable to model the probability of failure of these bulbs by an exponential density function with = 1000. Use this model to find the probability that a bulb fails within the first 200 hours. burns for more than 800 hours.

What is the median lifetime of these light bulbs?

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Example 4 – pg. 574 #12 According to the National Health Survey, the heights

of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches.

What is the probability that an adult male chosen at random is between 65 inches and 73 inches tall?

What percentage of the adult male population is more than 6 feet tall?

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Example 5 – pg. 574 #14 Boxes are labeled as containing 500 g of cereal. The

machine filling the boxes produces weights that are normally distributed with standard deviation of 12 g.

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• If the target weight is 500 g, what is the probability that the machine produces a box with less than 480 g of cereal?

• Suppose a law states that no more than 5% of a manufacturer’s cereal boxes can contain less than the stated weight of 500 g. At what target weight should the manufacturer set its filling machine?

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Book Resources

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Video Examples Example 2 – pg. 569 Example 4 – pg. 571 Example 5 – pg. 572

More Videos Expected values or means Calculating Probability

Wolfram Demonstrations Area of a Normal Distribution

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Web Resources

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http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/moreApps/gaussian.html

http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/moreApps/gaussian.html

http://www.youtube.com/watch?v=szjL60gAweE&feature=youtu.be