Question

1.The waiting time, in hour, between successive speeders spotted by a radar units is a continuous random variable with cumulative distribution function

f(x)={(1-e^(-8x), if x>0,0 otherwise)

 derive the characteristic function of x and use it to find the mean of x

Answer 1

The characteristic function has the form: $\varphi(t)=E[e^{itX}]$. We receive the following expression: $\varphi(t)=\int_{-\infty}^{+\infty}e^{itx}p(x)dx$, where $t\in{\mathbb{R}}$. $p(x)$ is the probability density function. $p(x)=f'(x)=8e^{-8x}$, $x>0$. For $x<0$ the probability density function is . We get: $\varphi(t)=8\int_{0}^{+\infty}e^{itx-8x}dx=\frac{8}{it-8}e^{itx-8x}|_0^{+\infty}=-\frac{8}{it-8}$. It is known that: $\frac{\partial\varphi(t)}{\partial t}|_{t=0}=iE[X]$. We get: $\frac{\partial\varphi(t)}{\partial t}=\frac{8i}{(it-8)^2}$. We receive: $E[X]=\frac18$.

Answer: $E[X]=\frac18.$