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Solved: Uniform Distribution The random-number generator
Chapter 7, Problem 17AYU(choose chapter or problem)
Problem 17AYU
Uniform Distribution The random-number generator on calculators randomly generates a number between 0 and 1. The random variable X, the number generated, follows a uniform probability distribution.
(a) Draw the graph of the uniform density function.
(b) What is the probability of generating a number between 0 and 0.2?
(c) What is the probability of generating a number between 0.25 and 0.6?
(d) What is the probability of generating a number greater than 0.95?
(e) Use your calculator or statistical software to randomly generate 200 numbers between 0 and 1. What proportion of the numbers are between 0 and 0.2? Compare the result with part (b).
Questions & Answers
QUESTION:
Problem 17AYU
Uniform Distribution The random-number generator on calculators randomly generates a number between 0 and 1. The random variable X, the number generated, follows a uniform probability distribution.
(a) Draw the graph of the uniform density function.
(b) What is the probability of generating a number between 0 and 0.2?
(c) What is the probability of generating a number between 0.25 and 0.6?
(d) What is the probability of generating a number greater than 0.95?
(e) Use your calculator or statistical software to randomly generate 200 numbers between 0 and 1. What proportion of the numbers are between 0 and 0.2? Compare the result with part (b).
ANSWER:
Answer:
Step 1
Uniform Distribution The random-number generator on calculators randomly generates a number between 0 and 1. The random variable X, the number generated, follows a uniform probability distribution.
(a) The graph of the uniform density function is as follows,
(b) We take X to be the angle at which the pointer comes to rest, so we use the interval [0, 1] = [a, b] as its range. Since all angles are equally likely, the probability density function should not depend on x and therefore should be constant. That is, we take f to be uniform f(x)=
The probability of generating a number between 0 and 0.2
Now, P(0 X
0.2)
P(cX
d)= Area of shaded rectangle =