IGNOU MTE 11 2018 - Suppose X has the density functionFind its moment generating f

Question

Suppose X has the density function

$f(x) = \left\{\begin{matrix} xe^-x,if x>0\\0, otherwise. \end{matrix}\right.$

Find its moment generating function whenever it exists.

14 Feb 2018

Answer :

Word Count : 371

To find the moment generating function (MGF) of the given probability density function (PDF) for $ X $, we start by recalling that the moment generating function is defined as: \[ M_X(t) = \mathbb{E}[e^{tX}] = \int_{-\infty}^{\infty} e^{tx} f(x) \, dx \] Given the PDF: \[ f(x) = \begin{cases} x e^{-x}, & x > 0 \\ 0, & \text{otherwise} \end{cases} \] We need to compute the MGF for $ t $, where $ t $ is any real number. Since the density is defined for $ x > 0 $, the integral reduces to: \[ M_X(t) = \int_{0}^{\infty} e^{tx} x e^{-x} \, dx \] This simplifies to: \[ M_X(t) = \int_{0}^{\infty} x e^{(t - 1)x} \, dx \] For the integral to converge, we require $ t - 1 < 0 $, or equivalently \( t < 1

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x	0	1	2	3	4	5	6	7	8
f	1	9	26	59	72	52	29	7	1