Question

Let X represent the time it takes from when someone enters the line for a roller coaster until they exit on the other side. Consider the probability model defined by the cumulative distribution function given below.
0 x < 3
F(x) = (x-3)/1.13 3 < x < 4.13
1 x > 4.13
A. What is E(X)? Give your answer to three decimal places.
B. What is the value c such that P(X < c) = 0.75? Give your answer to four decimal places.
C. What is the probability that X falls within 0.28 minutes of its mean? Give your answer to four decimal places.

Answer 1

Answer:

a) E(x)=3.565

b) c=3.8475 --> P(X < 3.8475) = 0.75

c) The probability that X falls above or below 0.28 min from the mean is P=0.4954.

Step-by-step explanation:

We have the cumulative distribution function as information.

a) To calculate the expected value, we can calculate the value of x in which F(x) equals 0.5. This happens for x=3.565.

$F(x)=\frac{x-3}{1.13} =0.5\\\\x-3=0.5*1.13=0.565\\\\x=0.565+3=3.565$

b) What is the value c such that P(X < c) = 0.75?

In this case, we have to calculate x to have F(x)=0.75

$F(x)=\frac{x-3}{1.13} =0.75\\\\x-3=0.75*1.13=0.8475\\\\x=0.8475+3=3.8475$

This happens for x=3.8475.

c) We have to calculate the probability that X falls above or below 0.28 min from the mean (x=3.565).

This is the probability that the time is between 3.285 and 3.845

$x_1=3.565-0.28=3.285\\\\x_2=3.565+0.28=3.845$

We can calculate this as:

$P(3.285<x<3.845)=F(3.845)-F(3.285)\\\\F(3.845)=\frac{3.845-3}{1.13}=\frac{0.845}{1.13}= 0.7478\\\\F(3.285)=\frac{3.285-3}{1.13}=\frac{0.285}{1.13}= 0.2522\\\\\\P(3.285<x<3.845)=F(3.845)-F(3.285)=0.7478-0.2522=0.4956$

The probability that X falls above or below 0.28 min from the mean is P=0.4954.