Question

g (12 points) The time between incoming phone calls at a call center is a random variable with exponential density p(x) = 1 r e −x/r on [0, [infinity]), where r = 20 ln(2). a. Verify that the function p(x) is a Probability Density Function.

Answer 1

`(1)p(x)\geq 0\\(2)\int_(0)^(\infty) p(x) dx=0`

Step-by-step explanation:

A function f(x) is a Probability Density Function if it satisfies the following conditions:

`(1)f(x)\geq 0\\(2)\int_(0)^(\infty) f(x) dx=0`

Given the function:

`p(x)=(1)/(r)e^(-x/r) \: on\: [0,\infty), where\:r=(20)/(ln(2))`

(1)p(x) is greater than zero since the range of exponents of the Euler's number will lie in
`[0,\infty).`

(2)

`\int_(0)^(\infty) p(x)=\int_(0)^(\infty) (1)/(r)e^(-x/r)\\=(1)/(r) \int_(0)^(\infty) e^(-x/r)\\=-(r)/(r)\left[e^(-x/r)\right]_(0)^(\infty)\\=-\left[e^(-\infty/r)-e^(-0/r)\right]\\=-e^(-\infty)+e^(-0)\\SInce \: e^(-\infty) \rightarrow 0\\e^(-0)=1\\\int_(0)^(\infty) p(x)=1`

The function p(x) satisfies the conditions for a probability density function.