14
votes
Accepted
Why must a product of symmetric random variables be symmetric?
To say that a random variable $W$ "has a symmetric distribution around zero" is saying that $W$ and $-W$ have the same distribution.
Let $X$ be another random variable and set $Y=WX.$ By ...
- 306k
10
votes
Accepted
Density of $|t_1 - t_2|$ where $t_1$ and $t_2$ are iid with $P(t) = \alpha e^{-t\alpha}$
It's surprising but correct. The exponential distribution is memoryless, meaning that the distribution of time until a decay is the same whenever you start. It's easy to show it's the same if you ...
- 28.9k
6
votes
Is the variance of the mean of a set of independent random variables equal to the average of their respective variances?
Given a set of random variables $X_1,\dots,X_n$, if they are independent, then
\begin{align}
\text{Var}(\overline X) &= \text{Var}\left(\frac{1}{n} \sum_{i=1}^n X_i\right) \\
&= \frac{1}{n^2}\...
- 3,658
6
votes
Accepted
Radial axis transformation in polar kernel density estimate
Consider any density $f$ for the circular parameter $\theta.$ The relevant integrals are of the form $$\Pr(\mathcal A) = \int_\mathcal{A}f(\theta)\,\mathrm d\theta$$ where $\mathcal A\subset[0,2\pi)$ ...
- 306k
5
votes
Density of $|t_1 - t_2|$ where $t_1$ and $t_2$ are iid with $P(t) = \alpha e^{-t\alpha}$
I would handle with a different approach which, imo, is more insightful:
Consider $T_i\overset{\textrm{iid}}{\sim}\mathrm{Exp}(\alpha), ~i\in\{1, 2\}; Z:=|T_1-T_2|.$ Now
\begin{align}\mathbb P(Z\leq z)...
- 5,137
4
votes
Existence of distribution that its difference of two iid RVs becomes a desired distribution
This is not the case if the two variables from the second distribution are independent.
For example, the uniform distribution over $[-1,1]$ cannot be expressed as the difference of two i.i.d. random ...
3
votes
Density of $|t_1 - t_2|$ where $t_1$ and $t_2$ are iid with $P(t) = \alpha e^{-t\alpha}$
For exponential distribution family, operating its survival function $S(t) = e^{-\alpha t}$ is usually slightly more convenient than treating the distribution function $F(t) = 1 - e^{-\alpha t}$. ...
- 10.4k
3
votes
Accepted
Variance of the difference of two iid sample means
The sample average $\bar X$ has expected value $\mu_1$ and variance $\sigma^2_1/n$. By the same token, $\bar Y$ has expected value $\mu_2$ and variance $\sigma^2_2/n$. Now using the linearity of ...
- 8,101
2
votes
Case when random variable X and its square $X^2$ are independent
I want to add that a sufficient and necessary condition for $X$ and $X^2$ are independent is that $X^2$ is degenerate.
The sufficiency is trivial. Conversely, suppose $X$ and $X^2$ are independent. ...
- 10.4k
2
votes
Case when random variable X and its square $X^2$ are independent
A cool answer is that $X$ does not need to be degenerate. Indeed, take $X$ to have the following.
$$
P(X = 1) = p\\
P(X = -1) = 1-p\\
P(X\notin\{-1,1\}) = 0\\
p\in(0,1)
$$
Then $P(X^2 = 1)= 1$ and $P(...
- 46.6k
1
vote
Calculating the n-th moment of a RV, including negative fractional moments
You can work this out by evaluating the integral directly.
$$
\mathbb{E}(X^n) = \int_{-\infty}^{\infty} x^n f_X(x)dx =
\int_{a}^{b} x^n \frac{1}{b-a}dx = \ldots
$$
As long as the interval $[a,b]$ ...
- 1,303
1
vote
What is the definition of the geometric mean of a random variable?
I have no idea what the "official" answer is, if any, and this might just be a repeat of Glen_b's answer with more calculus-y language (I don't know), but the whole idea is to be analogous ...
- 111
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