Linked Questions

2
votes
0answers
79 views

The $k$th moments of normal distribution [duplicate]

How to proof that the $k$th moments of random variable $X\backsim N(0, \sigma^2)$ which is: $$ E(X^k) = \begin{cases} \sigma^k(k!!) & \text{for k even;}\\ \ 0 & \text{for k odd} \end{cases} $...
2
votes
2answers
4k views

Rationale for $E[Z^4]=3$

Given $Z$ is a standard normal random variable with mean 0 and variance 1 ($Z \sim N(0,1)$), could anyone provide an explanation for why $E\left[ Z^4 \right] = 3$? I know that: $$ \begin{aligned} E \...
4
votes
3answers
350 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 ...
0
votes
0answers
2k 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?
2
votes
1answer
813 views

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 ...
3
votes
1answer
277 views

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 $X$ ...
4
votes
1answer
110 views

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 ...
2
votes
0answers
472 views

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
0answers
398 views

How to calculate E(1/Y) when Y is Inverse Gaussian distributed?

The Inverse Gaussian Distribution density is : $$\frac{\phi^{\frac{1}{2}}}{\sqrt{2\pi y^3}} exp[\frac{-\phi(y - \mu)^2}{2\mu^2y}]$$ Got to this integral: $$\int_0^\infty \frac{1}{y} \frac{\phi^{\frac{...
1
vote
0answers
357 views

MVUE and Cramer-Rao Lower Bound [duplicate]

Let $(X_1,...,X_n)$ be a random sample from a normal distribution $N(\mu,1)$. $T=\frac{1}{n}\sum_{i=1}^{n}X_{i}^2-1$ is an unbiased estimator for $\mu^2$ since : $$ E(T)=E(\frac{1}{n}\sum_{i=1}^{n}X_{...
1
vote
1answer
130 views

Stationarity of the TGARCH

I'm going through "GARCH models" by Francq and Zakoian (2010). They define the TGARCH(1,1) as $$\sigma_t = \omega + \beta_1 \sigma_{t-1} + \alpha_{1,+}\epsilon_{t-1}^+ - \alpha_{1,-}\epsilon_{t-1}^- $$...
0
votes
1answer
33 views

Choose one of two normal distribution that will give the probability of biggest value when sampling it

Suppose you have two (or more) normal distributions with different mean and variance. You can draw only one sample of only one of the available distributions. Your goal is to get the biggest value ...