# 5.2 The uniform distribution  (Page 3/16)

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## Try it

Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. Let X = the time, in minutes, it takes a student to finish a quiz. Then X ~ U (6, 15).

Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. Then find the probability that a different student needs at least eight minutes to finish the quiz given that she has already taken more than seven minutes.

P ( x >8) = 0.7778

P ( x >8 | x>7) = 0.875

Ace Heating and Air Conditioning Service finds that the amount of time a repairman needs to fix a furnace is uniformly distributed between 1.5 and four hours. Let x = the time needed to fix a furnace. Then x ~ U (1.5, 4).

1. Find the probability that a randomly selected furnace repair requires more than two hours.
2. Find the probability that a randomly selected furnace repair requires less than three hours.
3. Find the 30 th percentile of furnace repair times.
4. The longest 25% of furnace repair times take at least how long? (In other words: find the minimum time for the longest 25% of repair times.) What percentile does this represent?
5. Find the mean and standard deviation

e. $\mu =\frac{a+b}{2}$ and $\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}$
$((\mu , \frac{1.5+4}{2}), 2.75)$ hours and $\sigma =\sqrt{\frac{{\left(4–1.5\right)}^{2}}{12}}=0.7217$ hours

## Try it

The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. Let X = the time needed to change the oil on a car.

1. Write the random variable X in words. X = __________________.
2. Write the distribution.
3. Graph the distribution.
4. Find P ( x >19).
5. Find the 50 th percentile.
1. Let X = the time needed to change the oil in a car.
2. X ~ U (11, 21).
3. P ( x >19) = 0.2
4. the 50 th percentile is 16 minutes.

## Chapter review

If X has a uniform distribution where a < x < b or a x b , then X takes on values between a and b (may include a and b ). All values x are equally likely. We write X U ( a , b ). The mean of X is $\mu =\frac{a+b}{2}$ . The standard deviation of X is $\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}$ . The probability density function of X is $f\left(x\right)=\frac{1}{b-a}$ for a x b . The cumulative distribution function of X is P ( X x ) = $\frac{x-a}{b-a}$ . X is continuous.

The probability P ( c < X < d ) may be found by computing the area under f ( x ), between c and d . Since the corresponding area is a rectangle, the area may be found simply by multiplying the width and the height.

## Formula review

X = a real number between a and b (in some instances, X can take on the values a and b ). a = smallest X ; b = largest X

X ~ U (a, b)

The mean is $\mu =\frac{a+b}{2}$

The standard deviation is

Probability density function: $f\left(x\right)=\frac{1}{b-a}$ for $a\le X\le b$

Area to the Left of x : P ( X < x ) = ( x a ) $\left(\frac{1}{b-a}\right)$

Area to the Right of x : P ( X > x ) = ( b x ) $\left(\frac{1}{b-a}\right)$

Area Between c and d : P ( c < x < d ) = (base)(height) = ( d c ) $\left(\frac{1}{b-a}\right)$

Uniform: X ~ U ( a , b ) where a < x < b

• pdf: $f\left(x\right)=\frac{1}{b-a}$ for a ≤ x ≤ b
• cdf: P ( X x ) = $\frac{x-a}{b-a}$
• mean µ = $\frac{a+b}{2}$
• standard deviation σ $=\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}$
• P ( c < X < d ) = ( d c ) $\left(\frac{1}{b–a}\right)$

## References

McDougall, John A. The McDougall Program for Maximum Weight Loss. Plume, 1995.

Use the following information to answer the next ten questions. The data that follow are the square footage (in 1,000 feet squared) of 28 homes.

 1.5 2.4 3.6 2.6 1.6 2.4 2 3.5 2.5 1.8 2.4 2.5 3.5 4 2.6 1.6 2.2 1.8 3.8 2.5 1.5 2.8 1.8 4.5 1.9 1.9 3.1 1.6

The sample mean = 2.50 and the sample standard deviation = 0.8302.

The distribution can be written as X ~ U (1.5, 4.5).

What type of distribution is this?

In this distribution, outcomes are equally likely. What does this mean?

It means that the value of x is just as likely to be any number between 1.5 and 4.5.

What is the height of f ( x ) for the continuous probability distribution?

What are the constraints for the values of x ?

1.5 ≤ x ≤ 4.5

Graph P (2< x <3).

What is P (2< x <3)?

0.3333

What is P (x<3.5| x <4)?

What is P ( x = 1.5)?

zero

What is the 90 th percentile of square footage for homes?

Find the probability that a randomly selected home has more than 3,000 square feet given that you already know the house has more than 2,000 square feet.

0.6

Use the following information to answer the next eight exercises. A distribution is given as X ~ U (0, 12).

What is a ? What does it represent?

What is b ? What does it represent?

b is 12, and it represents the highest value of x .

What is the probability density function?

What is the theoretical mean?

six

What is the theoretical standard deviation?

Draw the graph of the distribution for P ( x >9).

Find P ( x >9).

Find the 40 th percentile.

4.8

Use the following information to answer the next eleven exercises. The age of cars in the staff parking lot of a suburban college is uniformly distributed from six months (0.5 years) to 9.5 years.

What is being measured here?

In words, define the random variable X .

X = The age (in years) of cars in the staff parking lot

Are the data discrete or continuous?

The interval of values for x is ______.

0.5 to 9.5

The distribution for X is ______.

Write the probability density function.

f ( x ) = $\frac{1}{9}$ where x is between 0.5 and 9.5, inclusive.

Graph the probability distribution.

1. Sketch the graph of the probability distribution.
2. Identify the following values:
1. Lowest value for $\overline{x}$ : _______
2. Highest value for $\overline{x}$ : _______
3. Height of the rectangle: _______
4. Label for x -axis (words): _______
5. Label for y -axis (words): _______

Find the average age of the cars in the lot.

μ = 5

Find the probability that a randomly chosen car in the lot was less than four years old.

1. Sketch the graph, and shade the area of interest.
2. Find the probability. P ( x <4) = _______

Considering only the cars less than 7.5 years old, find the probability that a randomly chosen car in the lot was less than four years old.

1. Sketch the graph, shade the area of interest.
2. Find the probability. P ( x <4| x <7.5) = _______
1. Check student’s solution.
2. $\frac{3.5}{7}$

What has changed in the previous two problems that made the solutions different?

Find the third quartile of ages of cars in the lot. This means you will have to find the value such that $\frac{3}{4}$ , or 75%, of the cars are at most (less than or equal to) that age.

1. Sketch the graph, and shade the area of interest.
2. Find the value k such that P ( x < k ) = 0.75.
3. The third quartile is _______
1. Check student's solution.
2. k = 7.25
3. 7.25

frequency destribution
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