Questions tagged [moment-generating-functions]

41 questions
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Proof that if $Z$ is standard normal, then $Z^2$ is distributed Chi-Square (1).

Suppose that $Z\sim N(0,1)$ and let $V=Z^2$. Prove that $V\sim \chi^2(1)$. I want to use the method of moment generating functions, because I already understand the proof using the method of ...
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Finding the moment generating function of the product of two standard normal distributions

The following question is on my homework assignment that I cannot figure out: Let U and V be independent random variables, each having a normal distribution with mean zero and variance one. Find ...
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Finding the M.G.F of product of two random variables.

We are given two independent standard normal random variables $X$ and $Y$. We need to find out the M.G.F of $XY$. I tried as follows : \begin{align} M_{XY}(t)&=E\left(e^{(XY)t}\right)\\&=\...
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Distribution of $Z$ from Moment Generating Function

Suppose that $X_1, X_2, ..., X_n$ are independent and identically distributed Exp(λ) random variables and let $Z = X_1 + X_2 + · · · + X_n$. Determine $M_Z(θ)$, the moment generating function of $Z$ ...
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Given a joint moment generating function, find $P(X<Y)$

Let random variables $X$ and $Y$ have joint MGF: $$M(t_1,t_2) = 1/2e^{t_1+t_2} + 1/4e^{2t_1+t_2} + 1/12e^{t_2} + 1/6e^{4t_1+3t_2}$$ I now need to find $P(X<Y)$. I know how to find the moments as I ...
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Finding Variance from a joint moment generating function

The random vars X and Y have, for all real values of $T_1, T_2$, the joint mgf $M(T_1 , T_2) = \frac{1}{2} e^{T_1 +T_2} + \frac{1}{4} e^{2T_1 +T2} + \frac{1}{12}e^{T_2} + \frac{1}{6} e^{4T_1 +3T_2}$ ...
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Moment generating function (MGF) of the ratio distribution $\displaystyle\frac{X}{Y}$

If we know the moment generating functions (MGFs) of the random variables $X$ and $Y$ to be $M_{X}(s)$ and $M_{Y}(s)$, respectively. The MGF of the sum $X+Y$ will $M_{X}(s) \cdot M_{Y}(s)$. So what ...
Let $Y_{i}$ be IID with the property $P(Y_{i}=1)=\frac{1}{2}$ and $P(Y_{i}=-1)=\frac{1}{2}$ then if $T=\min\{n:K_n=-c$ or $K_n =c\}$ with $K_n=\sum_{i=1}^{n}Y_{i}$, is a stopping time. Find the Moment ...
Is this statement correct: $\mathbb{E}[(X_1-\mu)^4]$ of $X_1 \sim N(\mu,\sigma^2)$ is the same as $\mathbb{E}[X_2^4]$ of $X_2 \sim N(0,\sigma^2)$? If so is there an easy way to show this? I used ...