Questions tagged [moment-generating-functions]

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Random variable $V$ has moment generating function $M(t)=e^{3e^{t}-1}$. What is $E(V)$? Note: you do not have to derive the mean.

Random variable $V$ has moment generating function $M(t)=e^{3e^{t}-1}$. What is $E(V)$? Note: you do not have to derive the mean. I have tried deriving the function , which i got $2e^2$ after ...
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2answers
132 views

Find the MGF of X

If $f(x) = (k + 1)x^2$ for $0 < x < 1$. Find the moment generating function of $X$. Do i do the integral from o to 1 of the above function?
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1answer
274 views

Finding probabilities of sum of independent random variables from their Moment Generating Function

This is a problem from A First Course in Probability, Sheldon Ross Ed. 7 Problem 7.75. I am really stumped on this one. The MGF of X is given by $M_X(t)=exp(2e^t-2)$ and the MGF of Y is $M_Y(t)= (\...
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2answers
131 views

What are the steps to take the derivative of this function?

This calculus derivation is giving me an extremely difficult time. I am having trouble understanding how to manipulate the $e^t$. The function is $$P*{e}^t/[(1-q)*{e}^t]$$ I want to take the ...
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1answer
205 views

What is the moment generating function and variance

Find the moments of the random variable $X$ if its moment generating function is $$M_X(t) = (1−p_1 −p_2)+p_1e^t +p_2e^{2t}.$$ What is the variance of $X$?
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0answers
481 views

Derive probability density function from moment generating function

I am given this information: $\tau$ is a constant, $\infty > \tau >0$ Let $X \sim G(n, \lambda)$, where $n \in \{1,2,...,\}$ ($G$ is referring to gamma distribution) Let $Y = n \tau + X$ ...
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4answers
2k views

What is the moment generating function from a density of a continuous random variable?

Let X be a random variable with probability density function $$f(x)=\begin{cases}xe^{-x} \quad \text{if } x>0\\0 \quad \text{ } Otherwise.\end{cases} $$ Determine the mgf of X whenever it exists. ...
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2answers
1k views

Find moment generating function of Y = $e^X$

Let $X$ ~ $N(0,1)$ and $Y=e^X$. Find the moment generating function of Y. I think I first need to find the cdf of Y. So I take: $F_Y(y) = P(Y \le y) = P(e^X \le y) = P(X \le ln(y)) = F_X(ln(y))$ I ...
2
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0answers
2k views

Moment-generating Function of a Continuous R.V. whose P.D.F is 1 from (0, 1)

I have been working on this problem for a few hours now and I feel I am missing something simple. The problem is to find the moment-generating function of a continuous random variable whose ...
0
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1answer
71 views

Moment Generating function of one random variable among multiple random variables

If the moment generating function of $X_1, X_2, X_3$ is How to find the moment generating function of just X1? Need some guidance on how to start..
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0answers
778 views

Why the probability characteristic function is always exist but moment generation function is not always exist?

I know that the characteristic function is always exist and moment generation function is not always exist instantly but don't know exactly mathematically.
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1answer
422 views

Is the bijection of independent random variables, independent random variables?

I'm trying to prove the following: $$M_{X}(t)=\prod\limits_{i=1}^n M_{X_i}\left(\frac{t}{n}\right)$$ Where $X := \frac{1}{n} \sum\limits_{i=1}^n X_i $ and all the random variables are independent. ...
2
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2answers
1k views

Calculating $E[(X-E[X])^3]$ by mgf

Calculate by mgf $E[(X-E[X])^3]$ where a. $X\sim B(n,p)$ b.$X\sim N(\mu,\sigma)$ Before I begin I thought symbolizing $Y=X-E[X]$ and then I'd derivative $M_Y(t)$ three times substitute $t=0$...
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2answers
265 views

Fitting probability distributions based on moment generating functions

Say I have a random variable $X$ with mgf $M_X(t) = 1 + a_1t + a_2t^2 + a_3t^3 + \cdots $ and another random variable $Y$ with a probability distribution determined by two parameters $\theta_1$ and ...
3
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3answers
7k views

$X$ standard normal distribution, $E[X^k]=?$

I'm stuck with a homework problem where we are supposed to prove that the expected value $E[X^k]$, if $X$ has standard normal distribution, is equal to: $$E[X^{2k}]=\frac{(2k)!}{k!\cdot2^k}.$$ But I ...