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Asked Oct 15, 2019
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6. Consider the function Lx (t) = In (Mx (t)). Show that E(X) = L* (0) and Var(X) = L% (0).
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6. Consider the function Lx (t) = In (Mx (t)). Show that E(X) = L* (0) and Var(X) = L% (0).

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Expert Answer

Step 1

Introduction:

Here, MX (t) denotes the moment generating function (mgf) of the random variable X; thus, MX (t) = E (etX).

The quantity LX (t) is the natural logarithm of MX (t).

Remember that MX (t), LX (t) are functions of t, and not that of x. Thus, all the derivatives are done here with respect to t, and not with respect to x.

Step 2

Calculation to prove E (X) = L՜ (0):

First, calculate the first order derivative of LX (t) with respect to t. After obtaining the d...

L()-In[M (]
L(t)L(t)
dt
MT, (t)
Now
M0)E)
dt
= E
e
dt
= E(Xe)
As a result
E(Xe)
L()
E (e)
EXe
L(0)
E(XT)
E(X)
E(1)
. E(x) L (0)(Proved)
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L()-In[M (] L(t)L(t) dt MT, (t) Now M0)E) dt = E e dt = E(Xe) As a result E(Xe) L() E (e) EXe L(0) E(XT) E(X) E(1) . E(x) L (0)(Proved)

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