# Density Functions: Time takes students to finish aptitude test

1. The time it takes for a student to finish an aptitude test has the density function

f(x) = c(x-1) (2-x), 1<x<2

= 0, otherwise.

Here the unit of time is an hour.

a) Evaluate the Normalizing constant c.

b) Plot the density function

c) Calculate the probability that a student will finish the test in less than 75 minutes

d) Calculate the probability that a student will finish the test between 1.5 and 2 hours.

e) Make some comments about the plot of the density function.

2. For the density f(x) = 375/(x^4), for x>5 and zero otherwise.

a) Find the median

b) Find the 90th percentile

c) Find the 10th percentile

3. Let f(x) be the uniform density on [0,1] and let g(x) be the density on [0,1] given by g(x)=ce^x, where c is the normalizing constant. Let F, and G be the corresponding CDFs. Find the density function corresponding to the new CDF H(x) = F(x)G(x).

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#### Solution Preview

1. The time it takes for a student to finish an aptitude test has the density function

f(x) = c(x-1) (2-x), 1<x<2

= 0, otherwise.

Here the unit of time is an hour.

a) Evaluate the Normalizing constant c.

b) Plot the density function

c) Calculate the probability that a student will finish the test in less than 75 minutes

d) Calculate the probability that a student will finish the test between 1.5 and 2 hours.

e) Make some comments about the plot of the density function.

Solution

a) Since is a probability density function we have

That is,

That is,

That is,

That is,

That is,

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That is,

That is,

Therefore, ...

#### Solution Summary

The expert determines the time it takes for students to finish aptitude tests.