# Continuous Random Variables

## The Uniform Distribution

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data are inclusive or exclusive of endpoints.

### 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 
. The standard deviation of X is 
. The probability density function of X is 
 for a ≤ x ≤ b. The cumulative distribution function of X is P(X ≤ x) = 
. 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 

The standard deviation is 
 
Probability density function:

 for 

Area to the left of  P(X < x) = (x – a)
 
Area to the right of  P(X > x) = (b – x)
 
Area between  P(c < x < d) = (base)(height) = (d – c)
 
Uniform: X ~ U(a, b) where a < x < b
1. pdf: 

for a ≤ x ≤ b

2. cdf: P(X ≤ x) = 

3. mean µ = 

4. standard deviation σ 

5. P(c < X < d) = (d – c)


### References

McDougall, J. A. (1995). The McDougall program for maximum weight loss. New York: Plume
Use the following information to answer the next 10 questions. The data that follow are the square footage (in 1,000 feet squared) of 28 homes:
The sample mean = 2.50 and the sample standard deviation = 0.8302.
The distribution can be written as X ~ U(1.5, 4.5).
Use the following information to answer the next eight exercises. A distribution is given as X ~ U(0, 12).
Use the following information to answer the next 12 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.

### Homework

For each probability and percentile problem, draw the picture.
Use the following information to answer the next three exercises. The Sky Train from the terminal to the rental–car and long–term parking center is supposed to arrive every eight minutes. The waiting times for the train are known to follow a uniform distribution.