Download 7.1.1 Uniform Probability Distributions

Continuous Random Variables
Page 1
Uniform Probability Distributions
Let the random variable X denote the outcome of the rand function of a graphing calculator.
That is, X is the outcome when a point is selected at random from the interval (0,1).
The probability density function for this situation is given by U(0,1), where
1.2
0, if x ≤ 0

U (0,1) = 1, if 0 < x < 1
 0, if x ≥ 1

1
0.8
0.6
0.4
0.2
0
-1
-0.5
0
0.5
1
1.5
2
1. Verify that this is a probability density function.
2. Determine the probability of selecting a random number between 0.25 and 0.60.
3. Conduct a simulation of this event using the rand function 20 times. Are the results
consistent with your answer in item 2?
Robert A. Powers
University of Northern Colorado
Continuous Random Variables
Page 2
Suppose that you spin a dial marked by degrees as shown in the figure on
the right so that it comes to rest at a random position.
4. Determine a probability density function that models this situation.
5. Find the probability that the dial will land between 5º and 300º.
Uniform Probability Density Function
The uniform probability density function on the interval [a, b] is the constant function defined
by
for a ≤ x ≤ b, and
0 for x < a or x > b
Challenge Problem
Continuous random variable distributions are often presented as a cumulative frequency
distribution rather than as the probability density function. That is, let the cumulative frequency
distribution F(x) be the probability that the random variable X is less than x.
x
F ( x) = P ( X < x) =
∫ f (t )dt , where f(t) is the probability density function.
−∞
Find F(x) for uniform probability distributions.
Robert A. Powers
University of Northern Colorado