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### 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 ...
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### 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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### 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 $X$ ...
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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 ...
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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: $$\int_0^\infty \frac{1}{y} \frac{\phi^{\frac{... 0answers 357 views ### MVUE and Cramer-Rao Lower Bound [duplicate] Let (X_1,...,X_n) be a random sample from a normal distribution N(\mu,1). T=\frac{1}{n}\sum_{i=1}^{n}X_{i}^2-1 is an unbiased estimator for \mu^2 since :$$ E(T)=E(\frac{1}{n}\sum_{i=1}^{n}X_{...
I'm going through "GARCH models" by Francq and Zakoian (2010). They define the TGARCH(1,1) as $$\sigma_t = \omega + \beta_1 \sigma_{t-1} + \alpha_{1,+}\epsilon_{t-1}^+ - \alpha_{1,-}\epsilon_{t-1}^-$$...