L' Hopital Rule Problem and MGFs Statistics

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In summary, the conversation discusses a problem with a moment generating function and finding the expected value and variance. The attempted solution involves using the quotient rule and l'Hospital's rule, but there is disagreement with the answer from Wolfram. The conversation concludes with a suggestion to expand the numerator in powers of t instead. There is also a brief discussion about the correct spelling and pronunciation of l'Hospital.
  • #1
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Homework Statement


Hey all, been having some problem with the rule..technique looks right but doesn't agree with wolfram's calc answer. Doing a moment generating function problem.

M(t)=(e^5t-e^4t)/t, Find EX and VARX


Homework Equations


M'(0)=e(x) and so on..


The Attempt at a Solution


Okay, this is an indeterminate equation.. that's why I had to use this rule.. But my answer keeps getting 9 while wolfram says 9/2, so I need some help here.

Ok, using quotient rule, differentiating, I get (5e^5t-4e^4t)/t + (e^4t-e^5t)/t^2. Correct me if I am wrong.

I separated the latter part of the equation into ((e^4t-e^5t)/t)(1/t), which then equals to (-1)(1/t)

Therefore, I get (5e^5t-4e^4t-1)/t, which is indeterminate again, differentiating both nominators, i get 9. What have I done wrong?
 
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  • #2
thoradicus said:

Homework Statement


Hey all, been having some problem with the rule..technique looks right but doesn't agree with wolfram's calc answer. Doing a moment generating function problem.

M(t)=(e^5t-e^4t)/t, Find EX and VARX


Homework Equations


M'(0)=e(x) and so on..


The Attempt at a Solution


Okay, this is an indeterminate equation.. that's why I had to use this rule.. But my answer keeps getting 9 while wolfram says 9/2, so I need some help here.

Ok, using quotient rule, differentiating, I get (5e^5t-4e^4t)/t + (e^4t-e^5t)/t^2. Correct me if I am wrong.

I separated the latter part of the equation into ((e^4t-e^5t)/t)(1/t), which then equals to (-1)(1/t)

Therefore, I get (5e^5t-4e^4t-1)/t, which is indeterminate again, differentiating both nominators, i get 9. What have I done wrong?

Using l'Hospital's rule is much harder than what you need. Just expand the numerator in powers of t and see what you get.

BTW: the correct spelling is l'Hospital (yes, like the place you go to for medical help)---not l'Hopital. It is pronounced low-pee-tall but not spelled that way.
 
  • #3
Ray Vickson said:
Using l'Hospital's rule is much harder than what you need. Just expand the numerator in powers of t and see what you get.

BTW: the correct spelling is l'Hospital (yes, like the place you go to for medical help)---not l'Hopital. It is pronounced low-pee-tall but not spelled that way.

I stand corrected haha.

Can you elaborate abit on what you mean by expanding the numerator? Is it the expansion of the partial fractions?:confused:
 
  • #4
thoradicus said:
I stand corrected haha.

Can you elaborate abit on what you mean by expanding the numerator? Is it the expansion of the partial fractions?:confused:

I mean: write out the series expansion of exp(5t) - exp(4t) in powers of t.
 
  • #5
Oh are you talking about the Taylor's Series?
 
  • #6
thoradicus said:
Oh are you talking about the Taylor's Series?

Don't keep asking---just go ahead and do it.
 
  • #7
Ray Vickson said:
BTW: the correct spelling is l'Hospital (yes, like the place you go to for medical help)---not l'Hopital. It is pronounced low-pee-tall but not spelled that way.

I would say if the French can drop pronounciation of consonants and replace them with circumflex accents then it's perfectly ok for English speakers to replace that with l'Hopital. Since we don't use those accents. This is really fussy. It reflects the current pronunciation better.
 

1. What is L'Hopital's rule?

L'Hopital's rule is a mathematical theorem used to evaluate limits of indeterminate forms. It states that if the limit of a function can be expressed as the ratio of two functions, and both functions have a limit of 0 or infinity, then the limit of the original function is equal to the limit of the ratio of the derivatives of the two functions.

2. How is L'Hopital's rule applied to problems?

L'Hopital's rule is typically applied to problems where we are trying to evaluate a limit that results in an indeterminate form, such as 0/0 or infinity/infinity. By taking the derivatives of the numerator and denominator and evaluating the resulting limit, we can often simplify the original problem and find the true value of the limit.

3. What is the connection between L'Hopital's rule and MGFs in statistics?

L'Hopital's rule can be used to simplify the calculation of moment generating functions (MGFs) in statistics. MGFs are used to find the moments of a probability distribution, and they often involve limits that result in indeterminate forms. By applying L'Hopital's rule, we can simplify these limits and find the MGF more easily.

4. What are some common applications of MGFs in statistics?

MGFs are commonly used in statistics to find the moments of a probability distribution, such as the mean and variance. They are also used to prove the Central Limit Theorem, which states that the sum of a large number of independent random variables will be approximately normally distributed.

5. Can L'Hopital's rule be applied to all types of limits?

No, L'Hopital's rule can only be applied to limits that result in indeterminate forms. It cannot be used to evaluate limits that result in a definite value, such as 1/2 or 5. Additionally, it is important to check the conditions for applying L'Hopital's rule, such as both functions having a limit of 0 or infinity, before using it to solve a problem.

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