Probability Function of X: Solutions & Steps

[tee yeh hun]

Let the probability function of X be given by
f(x)= c e^-x, x=1,2,3,...
(a)Find the value of c.
(b) Find the moment generating function of X,

Solutions (a) e-1
(b) (e-1) [ (e^t-1)/(1-e^t-1)]

Can anyone shows the steps?

 

[Stephen Tashi]

Let the probability function of X be given by
f(x)= c e^-x, x=1,2,3,...

It is more precise to say "probability density function".

(a)Find the value of c.

Solve the equation \sum_{x=0}^{\infty} Ce^{-x} = 1 for C.

C \sum_{x=0}^{\infty}e^{-x} = 1

(The sum is a geometric series with ratio e^{-1} )

Can you do the rest of the steps?

(b) Find the moment generating function of X,

M(t) is expected value of e^{tx}.

M(t) = \sum_{x=0}^\infty ( e^{tx} C e^{-x })

= C \sum_{x=0}^\infty (e^{t-1})^x

The sum is a geometric series with ratio e^{t-1}

Solutions (a) e-1

I got C = \frac{1}{1 - e^{-1}}

Is that the same thing?.

(b) (e-1) [ (e^t-1)/(1-e^t-1)]

I got \frac{C}{ 1 - e^{t-1}}

 

[tee yeh hun]

thank you for the showings, appreciate it. Yes i can do the rest of it.

 

[tee yeh hun]

I got C = \frac{1}{1 - e^{-1}}

Is that the same thing?.

Here it goes,
It stated that the domain of x is from 1 to infinity(I am not sure why they didn't include 1.1, 2.3 , 5.89 these kind of numbers)

But unfortunately, if we are discussing probability density function, we use integral to find out where are the area covers.

∫Ce^-xdx = 1 [1,infinity)

the integral or summation method is the same idea actually.

Then we will find out that our C is which is 1/(e^1)

but for all no reason the answer of (a) is e-1 not e the power of -1

 

[tee yeh hun]

the whole question is as follow
Let the probability function of X be given by
f (x) = ce-x , x = 1, 2, 3, ….
(a) Find the value of c.
(b) Find the moment generating function of X.
(c) Use the result obtained from (b) to find E(X).
(d) Find the probability generating function of X.
(e) Verify that E(X) obtained using probability generating function is same as in (c).


Answers:

 

[Stephen Tashi]

Perhaps there is a typographical error in the problem. Perhaps there is more than one typographical error.