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

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

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!