Random Variables and Probability Distribution

Question 3.32

The proportion of the budgets for a certain type of industrial company that is allocated to environmental and pollution control is coming under scrutiny. A data collection project determines that the distribution of these proportions is given by:

f (y) = 5 (1-y)^4,  0<=y<=1
          0, elsewhere

a) Verify that the above is a valid density
b) What is the probability that a company chosen at random expends less than 10% of its budget on environmental and pollution controls ?
c) What is the probability that a company selected at random spends more than 50% on environmental and pollution control ?




Solution:
a)


Let u = 1-y


















Thus, this is a valid density function.

b)  

c)