Question

Verify the formula for the moment generating function of a uniform random variable that is given in Table 7.2. Also, differentiate to verify the formulas for the mean and variance.

Short answer

It has been verify the formula for the moment generating function of a uniform random variable that is given in Table as$\frac{(b-a{)}^2}{12}$.

Step-by-step solution

Given information

The moment generating function of a uniform random variable that is given in Table

Solution

If $X~U(a,b)$

Then $M_x\left(t\right)=E\left[e^{tx}\right]$

$={\int }_a^b e^{tx}\frac{1}{b-a}dx$

$=\frac{1}{b-a}{\left[\frac{e^{tx}}{t}\right]}_a^b$

$=\frac{e^{bt}-e^{at}}{t(b-a)}$

Then it has been verified

Solution

Now,

${\mathrm{M}}_{\mathrm{x}}\left(\mathrm{t}\right)=\frac{1}{(b-a)}\left[\frac{e^{bt}-e^{at}}{t}\right]$

$=\frac{1}{(b-a)}\left[\frac{\left(1+bt+\frac{(bt{)}^2}{2!}+\frac{(bt{)}^3}{3!}+\dots \dots .\right)-\left(1+at+\frac{(at{)}^2}{2!}+\frac{(at{)}^3}{3!}+\dots \dots \right)}{t}\right]$

$=\frac{1}{(b-a)}\left[\frac{(b-a)t}{t}+\frac{\left(b^2-a^2\right)t^2}{2!t}+\frac{\left(b^3-a^3\right)t^3}{3!t}+\dots \dots \right]$

$=1+\frac{(b+a)}{2}t+\frac{\left(b^2+a^2+ab\right)}{6}{t}^2+\dots \dots \dots$

Solution

Now,

${\mathrm{M}}_{\mathrm{x}}^{\cdot }\left(t\right)=\frac{(b+a)}{2}+\frac{\left(b^2+a^2+ab\right)}{6}2t+\dots \dots \dots$

${\mathrm{M}}_{\mathrm{X}}^{\ast }\left(\mathrm{t}\right)=\frac{b^2+a^2+ab}{3}+o\left(t\right);o\left(t\right)=$Terms having higher power of t

So,

$\mathrm{E}\left(\mathrm{X}\right)={\left.{\mathrm{M}}_{\mathrm{X}}^{\mathrm{\prime }}\left(\mathrm{t}\right)\right|}_{t=0}=\frac{(b+a)}{2}$

$\mathrm{E}\left({\mathrm{X}}^2\right)={\left.{\mathrm{M}}_{\mathrm{X}}^{\ast }\left(t\right)\right|}_{t=0}=\frac{\left(b^2+a^2+ab\right)}{3}$

$\Rightarrow V\left(X\right)=E\left(X^2\right)-\left[E\right(X){]}^2$

$=\frac{b^2+a^2+ab}{3}-\frac{b^2+a^2+2ab}{4}$

$=\frac{4\left(b^2+a^2+ab\right)-3\left(b^2+a^2+2ab\right)}{12}$

$=\frac{(b-a{)}^2}{12}$

Final answer

It has been verify the formula for the moment generating function of a uniform random variable that is given in Table as$\frac{(b-a{)}^2}{12}$.