Questions tagged [moment-generating-function]
A description of a probability distribution which is related to the Laplace transform. Use also for its logarithm, the cumulant generating function.
287 questions
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Statistics – Linear Inference – What is the moment generating function of non-central t-distribution?
Non-central t-distribution, mgf.
What is the moment generating function of non-central t-distribution?
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Theoretical Applications of Combinants in Probability
I've recently come across the concept of combinants while reading about probability theory. The Wikipedia article on combinants provides a basic overview but doesn't go into much any detail about how ...
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Beta Distribution and the Moment Problem (citation needed)
As I understand it, the Beta Distribution is uniquely defined in terms of its moments (i.e. the Moment Problem has a unique solution on the values of its moments). The Wikipedia article of the Beta ...
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On the finiteness of moments of a distribution
Consider a continuous random variable $X\equiv\log(Y)$. Assume that
$$
E(\exp(\alpha X))< \infty \quad \text{ for some $\alpha>0$}
$$
I would like to understand what does this assumption imply ...
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Proving that mgf determines distribution via Laplace transform
I am reading this question and the answer provided there about the moment generating function (mgf) and how its uniqueness can be proved via the uniqueness of Laplace transforms. In my book, Measure ...
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what is the formula for calculation of fourth raw moment (or central moment) from variance and 3rd central moment (or raw moment)?
Basically the title. I can't seem to find any solution for this.
I have the mean, variance or the second central moment and third central moment and third raw moment. I need to find the fourth raw ...
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How to find probability from $E[X^n]$?
It is given that $E[X^n] = \frac{2}{5}(-1)^n + \frac{2^{n+1}}{5}+\frac{1}{5}$, where $n=1,2,3,\ldots.$
I need to find $P(|X-\frac{1}{2}| > 1)$.
What my approach is :
I have opened the modulus ...
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How do I compute a probability from the MGF?
I have a random variable $X$ with moment generating function:
$$m_X(t) = \frac{2}{9} + \frac{e^{-t}}{9} + \frac{e^{-2t}}{9} + \frac{2e^{t}}{9} + \frac{e^{2t}}{3}.$$
I want to find the probability $\...
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Zero variance but non-zero skewness
I was thinking of a hypothetical distribution where the mean(first cumulant) is non-zero, second cumulant(variance) is zero, and the third cumulant(skewness) is non-zero. The higher order cumulants ...
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How can we efficiently find the fourth moment of a Poisson distribution?
Suppose we have $X\sim \textrm{Poisson}(\lambda)$ and we know that moment generating function $M(t)=\mathbb{E}(e^{tX})$. How do we use the moment generating function property $M^k(0)=\mathbb{E}(X^k)$ ...
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Find PDF from approximated MGF
I have an array of values of MGF (it is evaluated at some points).
The plot of it is shown (blue curve): .
Is it possible to find PDF knowing MGF in such form?
I tried to fit MGF with some curve (you ...
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MGF of "generalised" Negative Binomial distribution
Would the "extended" Negative Binomial have the same MGF as Negative Binomial? (See the definition of "extended" Negative Binomial below by Wikipedia)
Could someone please help ...
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How to find the MGF of the max of a set of i.i.d. exponential random variables
As the title suggests, I would like to find the MGF of the max of iid exponential random variables. Assume $Z=\max(x_{1},...,x_{n})$, where $x_{i}$ is distributed as exponential($\beta$) and has pdf $\...
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Proving relation between counts and arrival times via transforms
I'll state what I'm trying to prove below.
For a Poisson process $N(t) \sim \operatorname{Poisson}(\lambda t)$,
$$
P\left(S_n \leq t\right)=P\left(N(t) \geq n\right)=1-P(N(t)<n),
$$
where $S_n=\...
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Chernoff Bounds for Independent Bernoulli Sums
What is wrong with this proof? Can you notice that? or I am wrong? In my opinion, in the R.H.S. of the inequality (3.2), the index of 'e' is negative but it must be positive if we use the given proof ...
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