# Tag Info

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
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

### 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
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

### 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

### 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 ...

### 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
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

### 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