Solve the below:
Q1. The time it takes for a student to finish an aptitude test has the density function
f(x) = c(x-1) (2-x), 1
f(x)= 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.
Q2. 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
Q3. 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).