1 vote
12k views

### Calculate moment generating function of normal distribution [duplicate]

The moment generating function of a normal distribution is defined as $M(t) = \int_{-\infty}^\infty e^{tx}\frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2} dx$ In a book I’m ...
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558 views

### Standard normal moments [duplicate]

My textbook says, without proof, that $E(X^4)=3$, where $X\sim N(0,1)$. Is it so simple to obtain?
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### Expected value of functions of Gaussian random variable

I am trying to compute the following expectations: (i) $E\left[\frac{1}{a^X}\right]$, (ii) $E\left[\frac{X^2}{a^X}\right]$, (iii) $E\left[\frac{X}{a^X}\right]$. In above, $a$ is a constant and $X$ ...
795 views

### Possible to use moment generating function of standard normal to find variance of noncentral $\chi^{2}$?

In a homework problem, I have been asked to find the variance of the noncentral $\chi_{n,\delta}^{2}$ distribution with degrees of freedom $n$ and noncentral parameter $\delta >0$ by using the ...
837 views

### How to compute this moment of a bivariate normal distribution?

Consider that $X \sim N(\mu_X, \sigma_X^2)$ and $Y \sim N(\mu_Y, \sigma^2_Y)$ and $\text{Cov}[X, Y] = \sigma_{XY}$ where $\sigma_{XY} \ne 0$. How can an expression for $\text{E}[Y(X-\mu_X)^p]$ in ...
3k views

### expected value of Brownian Motion

Suppose I have a brownian motion $B(t)$, how to calculate the Expected value of $B(t)$ to the power of any integer value $n$? Intuition told me should be all 0. But how to make this calculation?
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### Moment generating function of a (normally distributed) random variable

I am given 3 things: $Z$ follows a normal distribution $N(0,1)$ $Y=e^{X}$ $X=3-2Z$ What is the moment generation function of $X$ and the $r^{th}$ moment of $Y$ ($E[Y^{r}]$)? My attempt: I know ...
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### Help getting the variance of $Y$

We have $Z \sim N(0,1)$ and $Y=a+bZ+cZ^2$. What is the variance of $Y$? This is what I did: $$\mbox{Var} (Y)=0+b^2 \, \mbox{Var} (Z)+c^2 \, \mbox{Var} (Z^2) = b^2 + c^2 \, \mbox{Var} (Z^2)$$ To get ...
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### Fourth moment of complex Gaussian r.v

Let $x \sim \mathcal{CN}(0,a)$ (a complex Gaussian random variable) and I know that the p.d.f. of $z$ is defined by: $$p(z) = \frac{1}{\sqrt{2\pi a}} e^{-\frac{|z|^2}{2a}}$$. Given that, how can I ...
1 vote