Solve f(x) = ce^-x, Find E(x) and Probability Generating Function

In summary, the conversation discusses finding the value of c and the moment generating function of X using the given function. It also involves finding the expected value of X and the probability generating function of X. The conversation also mentions verifying the expected value using the probability generating function. The conversation includes a step-by-step explanation of how to find the value of c and provides guidance on how to continue with the moment generating function calculation.
  • #1
JoanneTan
3
0
Here is the question,

f(x) = ce^-x , x = 1, 2, 3...

Find the value of c.
Find the moment generating function of X.
Use the result obtained, find E(x).
Find the probability generating function of X.
Verify that E(x) obtained using probability generating function is same as the first E(x) founded.

I check the answer of the book, but it's wrong. Can someone help me? I'm looking for the working.
 
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  • #2
JoanneTan said:
Here is the question,

f(x) = ce^-x , x = 1, 2, 3...

Find the value of c.
Find the moment generating function of X.
Use the result obtained, find E(x).
Find the probability generating function of X.
Verify that E(x) obtained using probability generating function is same as the first E(x) founded.

I check the answer of the book, but it's wrong. Can someone help me? I'm looking for the working.

PF rules require that you show your work.
 
  • #3
Ok.. For the first question which is to find value of c.
ImageUploadedByPhysics Forums1398268199.770761.jpg

But the answer given is e - 1.
It's not e^-1.. I'm confusing how to get e - 1.
 
  • #4
Yes, you must have [itex]ce^{-1}+ ce^{-2}+ ce^{-3}+ \cdot\cdot\cdot= 1[/itex]

You can factor out [itex]ce^{-1}[/itex] and have [itex]ce^{-1}(1+ e^{-1}+ e^{-2}+ \cdot\cdot\cdot)[/itex]

That is, as you say, a geometric series with common factor [itex]e^{-1}[/itex] so is equal to [itex]\frac{ce^{-1}}{1- e^{-1}}= 1[/itex]. That, you have. Now multiply both numerator and denominator by [itex]e[/itex]:
[itex]\frac{c}{e- 1}= 1[/itex].
 
Last edited by a moderator:
  • #5
Oh! Ok.. I get u now.. Thanks a lot! But the second question, moment generating function,
As u can see from the photo, I done until half, can help me to continue? Cause dono how to substitute.
 
  • #6
JoanneTan said:
Oh! Ok.. I get u now.. Thanks a lot! But the second question, moment generating function,
As u can see from the photo, I done until half, can help me to continue? Cause dono how to substitute.

You need to calculate the sum [tex]\sum_{n=1}^{\infty} e^{-n} e^{kn}
= \sum_{n=1}^{\infty} r^n, \text{ where } r = e^{k-1}[/tex]
You have already seen how to do such summations; look at part (a)!
 

Related to Solve f(x) = ce^-x, Find E(x) and Probability Generating Function

1. What is f(x) and how is it related to e^-x?

F(x) is a mathematical function that represents the probability of a certain event occurring. In this case, it represents the probability of a random variable x having a certain value. The function e^-x is the natural exponential function, which is often used in probability calculations.

2. What is E(x) and how is it calculated?

E(x) is the expected value or mean of the random variable x. It represents the average value that would be obtained if the experiment or process was repeated a large number of times. It is calculated by multiplying each possible value of x by its corresponding probability and then summing all of these values.

3. What is a Probability Generating Function and how is it used?

A Probability Generating Function (PGF) is a mathematical function that is used to calculate the probabilities of a discrete random variable. It is defined as the sum of the probabilities of all possible values of the random variable multiplied by the corresponding powers of a variable t. It is useful in analyzing the behavior of a random variable and calculating its moments.

4. How do you solve f(x) = ce^-x for x?

To solve this equation, you can take the natural logarithm of both sides to eliminate the exponential term. This will result in ln(f(x)) = ln(c) - x. Then, you can solve for x by subtracting ln(c) from both sides and multiplying by -1.

5. Can f(x) be used to model real-life situations?

Yes, f(x) can be used to model real-life situations in which there is a discrete random variable. For example, it can be used to model the number of successes in a series of independent trials or the number of defects in a batch of products. However, it is important to note that the model is only as accurate as the assumptions and data used to create it.

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