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Imagine that a friend of yours is
Chapter 7, Problem 15AYU(choose chapter or problem)
Problem 15AYU
Imagine that a friend of yours is always late. Let the random variable X represent the time from when you are supposed to meet your friend until he shows up. Suppose your friend could be on time (x = 0) or up to 30 minutes late (x = 30), with all intervals of equal time between x = 0 and x = 30 being equally likely. For example, your friend is just as likely to be from 3 to 4 minutes late as he is to be 25 to 26 minutes late. The random variable X can be any value in the interval from 0 to 30, that is, 0 ≤ x ≤ 30. Because any two intervals of equal length between 0 and 30, inclusive, are equally likely, the random variable X is said to follow a uniform probability distribution.
Area as a Probability
Problem Refer to the situation in Example 1.
(a) What is the probability your friend will be between 10 and 20 minutes late?
(b) It is 10 a.m. There is a 20% probability your friend will arrive within the next ______
minutes.
Approach Use the graph of the density function in Figure 1 to find the solutions.
Solution
(a) We want to find the shaded area in Figure 2(a). The width of the shaded rectangle is and its height is The area between 10 and 20 is The probability your friend is between 10 and 20 minutes late is
(b) We are given the area of the shaded region in Figure 2(b). Here we need to determine the width of the rectangle so that its area is 0.2. We solve and find x0 = 30(0.2) = 6. There is a 20% probability your friend will arrive within the next 6 minutes, or by 10:06 a.m.
Find the probability that your friend is at least 20 minutes late.
Questions & Answers
QUESTION:
Problem 15AYU
Imagine that a friend of yours is always late. Let the random variable X represent the time from when you are supposed to meet your friend until he shows up. Suppose your friend could be on time (x = 0) or up to 30 minutes late (x = 30), with all intervals of equal time between x = 0 and x = 30 being equally likely. For example, your friend is just as likely to be from 3 to 4 minutes late as he is to be 25 to 26 minutes late. The random variable X can be any value in the interval from 0 to 30, that is, 0 ≤ x ≤ 30. Because any two intervals of equal length between 0 and 30, inclusive, are equally likely, the random variable X is said to follow a uniform probability distribution.
Area as a Probability
Problem Refer to the situation in Example 1.
(a) What is the probability your friend will be between 10 and 20 minutes late?
(b) It is 10 a.m. There is a 20% probability your friend will arrive within the next ______
minutes.
Approach Use the graph of the density function in Figure 1 to find the solutions.
Solution
(a) We want to find the shaded area in Figure 2(a). The width of the shaded rectangle is and its height is The area between 10 and 20 is The probability your friend is between 10 and 20 minutes late is
(b) We are given the area of the shaded region in Figure 2(b). Here we need to determine the width of the rectangle so that its area is 0.2. We solve and find x0 = 30(0.2) = 6. There is a 20% probability your friend will arrive within the next 6 minutes, or by 10:06 a.m.
Find the probability that your friend is at least 20 minutes late.
ANSWER:
Problem 15AYU
Answer:
Step1:
We have the random variable X represent the time from when you are supposed to meet your friend until he shows up. Suppose your friend could be on time (x = 0) or up to 30 minutes late
(x = 30), with all intervals of equal time between x = 0 and x = 30 being equally likely. For example, your friend is just as likely to be from 3 to 4 minutes late as he is to be 25 to 26 minutes late. The random variable X can be any value in the interval from 0 to 30, that is, 0 ≤ x ≤ 30. Because any two intervals of equal length between 0 and 30, inclusive, are equally likely, the random variable X is said to follow a uniform probability distribution.
a).
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 10 and 20 is
Now, P(cX
d) = P(10
X
20) and P(a
X
b) = P(0 ≤ x ≤ 30)
P(cX
d)= Area of shaded rectangle =