chapter 7.3 - probability distributions for continuous ... · probability distributions for...

AP STATISTICS

7.3 Probability Distributions for Continuous Random

Variables

7.3 Objectives:Ø Understand the definition and properties of continuous random variablesØ Be able to represent the probability distributions of continuous random variables using tables, formulae, or histograms

7.3 Objectives:Ø Be able to interpret formulaic or graphic representations of probability distributions of continuous random variables.Ø Be able to calculate probabilities such as P(x < a), P(x ≥ a), and P(a ≤ x ≤ b)

Let X = the weight (in pounds) of a full-term newborn child (reported to the nearest pound

x 4 5 6 7 8 9p(x) 0.26 0.31 0.21 0.13 0.06 0.03

Consider the random variable:x = the weight (in pounds) of a full-term

newborn child

What type of variable is this?

Consider the random variable:x = the weight (in pounds) of a full-term

newborn childSuppose that weight is reported to the nearest pound. The following probability histogram displays the distribution of weights.

The area of the rectangle centered over 7 pounds

represents the probability 6.5 < x < 7.5

Consider the random variable:x = the weight (in pounds) of a full-term

newborn childSuppose that weight is reported to the nearest pound. The following probability histogram displays the distribution of weights.

What is the sum of the areas of all the rectangles?

Consider the random variable:x = the weight (in pounds) of a full-term

newborn childSuppose that weight is reported to the nearest pound. The following probability histogram displays the distribution of weights.

Now suppose that weight is reported to the nearest 0.1 pound. This would be the probability histogram.

Consider the random variable:x = the weight (in pounds) of a full-term

newborn childSuppose that weight is reported to the nearest pound. The following probability histogram displays the distribution of weights.

Now suppose that weight is reported to the nearest 0.1 pound. This would be the probability histogram.

Notice that the rectangles are narrower and the histogram begins to

have a smoother appearance.

Consider the random variable:x = the weight (in pounds) of a full-term

newborn childSuppose that weight is reported to the nearest pound. The following probability histogram displays the distribution of weights.

Now suppose that weight is reported to the nearest 0.1 pound. This would be the probability histogram.

If weight is measured with greater and greater accuracy, the histogram

approaches a smooth curve.

Consider the random variable:x = the weight (in pounds) of a full-term

newborn childSuppose that weight is reported to the nearest pound. The following probability histogram displays the distribution of weights.

Now suppose that weight is reported to the nearest 0.1 pound. This would be the probability histogram.

This is an example of a density curve.

The shaded area represents the probability 6 < x < 8.

Probability Distributions for Continuous Variables

• Is specified by a curve called a density curve.

• The function that describes this curve is denoted by f(x) and is called the density function.

• The probability of observing a value in a particular interval is the area under the curve and above the given interval.

Properties of continuous probability distributions

1. f(x) > 0 (the curve cannot dip belowthe horizontal axis)

2. The total area under the density curve equals one.

Let X denote the amount of gravel sold (in tons) during a randomly selected week at a particular sales facility. Suppose that the density curve has a height f(x) above the value x, where

The density curve is shown in the figure:

⎩⎨⎧ ≤≤−

=otherwise0

10)1(2)(

xxxf

1

1

2

Tons

Density

1 – ½(0.5)(1) = .75

Gravel problem continued . . .What is the probability that at most ½ ton of gravel is sold during a randomly selected week?

1

1

2

Tons

Density

P(x < ½) =The probability would be the

shaded area under the curve and above the interval from 0 to 0.5.

1

1

2

Tons

Density

P(x = ½) =The probability would be the area

under the curve and above 0.5.

How do we find the area of a line segment?

Since a line segment has NO area, then the probability that exactly ½

ton is sold equals 0.

= 1 – ½(0.5)(1) = .75

Gravel problem continued . . .What is the probability that less than ½ ton of gravel is sold during a randomly selected week?

1

1

2

Tons

Density

P(x < ½) = P(x < ½)

Suppose x is a continuous random variable defined as the amount of time (in minutes) taken by a clerk to process a certain type of application form. Suppose x has a probability distribution with density function:

The following is the graph of f(x), the density curve:

⎩⎨⎧ <<

=otherwise0

645.)(

xxf

0.5

4 5 6Time (in minutes)

Den

sity

Application Problem Continued . . .What is the probability that it takes more than 5.5 minutes to process the application form?

0.5

4 5 6Time (in minutes)

Den

sity

P(x > 5.5) = .5(.5) = .25

Application Problem Continued . . .What is the probability that it takes more than 5.5 minutes to process the application form?

0.5

4 5 6Time (in minutes)

Den

sity

P(x > 5.5) = .5(.5) = .25When the density is constant over an interval (resulting in a horizontal density curve), the probability distribution is called a uniform

distribution.

Other Density CurvesSome density curves resemble the one below. Integral calculus is used to find the area under the these curves.Don’t worry – we will use tables (with the values already calculated). We can also use calculators or statistical software to find the area.

The probability that a continuous random variable x lies between a lower limit a and an upper limit b is

P(a < x < b) = (cumulative area to the left of b) –(cumulative area to the left of a)

P(a < x < b) = P(x < b) – P(x < a)

Let X be the amount of time (in minutes) that aparticular San Francisco commuter must wait for aBART train. Suppose that the density curve is as pictured (a uniform distribution):

a. What is the probability that x is less than 10 minutes? more than 15 minutes?

a. What is the probability that x is less than 10 minutes? more than 15 minutes?

b. What is the probability that X is between 7 and 12 minutes?

c. Find the value c for which P(x < c) = .9.

The graph gives the distribution of the yearly amount of rainfall in Rainy City:

In a randomly selected year,(a) What is the probability that Rainy City got more

than eight inches of rain?

P(Rainy City got more than 8 inches of rain)= P(X > 8) = 1 – P(X < 8) = 1 – [0.44 + 0.30 + 0.15 + 0.06] = 1 – 0.95 = 0.05

The graph gives the distribution of the yearly amount of rainfall in Rainy City:

In a randomly selected year,(b) What is the probability that Rainy City got between two and six inches of rain?

P(Rainy City got between 2 and 6 inches of rain)= P(2 < X < 6) = 0.30 + 0.15 = 0.45

The graph gives the distribution of the yearly amount of rainfall in Rainy City:

In a randomly selected year,(c) What is the probability that Rainy City got

exactly two inches of rain?P(Rainy City got exactly 2 inches of rain) = 0

The graph gives the distribution of the yearly amount of rainfall in Rainy City:

In a randomly selected year,(d) What is the probability that Rainy City got at most six inches of rain?P(Rainy City got at most 6 inches of rain)= P(X ≤ 6) = 0.44 + 0.30 + 0.15 = 0.89

7.3 Objectives:ü Understand the definition and properties of continuous random variablesü Be able to represent the probability distributions of continuous random variables using tables, formulae, or histograms

7.3 Objectives:ü Be able to interpret formulaic or graphic representations of probability distributions of continuous random variables.ü Be able to calculate probabilities such as P(x < a), P(x ≥ a), and P(a ≤ x ≤ b)

• HW: Read 7.3: Probability Distributions for Continuous Random Variables

For Tonight:

P414: 7.20, 7.21, 7.22, 7.23, 7.24, 7.26