The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous random variable with cumulative distribution function f(x) = 0, x< 0, 1 − e −8x, x≥ 0. find the probability of waiting less than 12 minutes between successive speeders (a) using the cumulative distribution function of x; (b) using the probability density function of x.

The probability of waiting less than 12 minutes between successive speeders using  cumulative distribution function of x and using the probability density function of x are respectively; 0.7981 and 0.7981

Solving Cumulative Probability Distributive Functions

The  cumulative distribution function of the random variable X, the waiting time, in hours, between successive speeders spotted by a radar unit is:

A) We want to find the probability of waiting less than 12 minutes between successive speeders. 12 minutes is also 12/60 hours = 0.2 hours

Using Cumulative distributive function, the probability is:

P(x < 20) = 1 - at x = 0.2

Thus;

P(x < 20) =

P(x < 20) = 0.7981

B) Using  probability density function of X is:

This gives;

P(x < 20) = between the boundaries of 0.2 and 0

Integrating gives;

P(x < 20) =

Solving this gives us;

P(x < 20) = 0.7981

Read more about cumulative probability distribution at; link

Given the CDF



we have PDF



Note that is wait-time given in hours, so we need to convert from minutes to hours:



so we're looking for .

The CDF gives us this value right away, since for any continuous random variable with distribution function :



To use the PDF, we need to integrate:

Looks like choice B would be the best description of an output.. Usually, output refers to the dependent value of a function rule.  It is also correct that the y coordinate of ordered pairs, graphed points, and in a function table.