Linked Questions
21 questions linked to/from How to calculate the expected value of a standard normal distribution?
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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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1
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558
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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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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:
$$
E(X^k) =
\begin{cases}
\sigma^k(k!!) & \text{for k even;}\\
\
0 & \text{for k odd}
\end{cases}
$...
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Problem with the expectation of transformed random variable [duplicate]
I read a paper where the random variable $z$ is assumed to follow a log-normal distribution.
\begin{equation}
\mathbb{E}\left(z^{1-\chi}\right)=\int_{0}^{\infty}z^{1-\chi}\frac{1}{z\sqrt{2\pi s^{2}}}e^...
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Second order moment of the Gaussian Distribution [duplicate]
In Bishops book, "Statistical Pattern Recognition", there is one exercise, which states to derive the second order moment of the Gaussian Distribution:
$E[x^2] = \int_{-\infty}^{\infty} N(x|...
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2
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Rationale for $E[Z^4]=3$
Given $Z$ is a standard normal random variable with mean 0 and variance 1 ($Z \sim N(0,1)$), could anyone provide an explanation for why $E\left[ Z^4 \right] = 3$?
I know that:
$$ \begin{aligned} E \...
- 2,630
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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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1
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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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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 ...
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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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2
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1
answer
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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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4
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1
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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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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 :
$$ E(T)=E(\frac{1}{n}\sum_{i=1}^{n}X_{...
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How to calculate E(1/Y) when Y is Inverse Gaussian distributed?
The Inverse Gaussian Distribution density is : $$\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{...
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