Timeline for Show that the sum of two random variables is a mixture
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 9, 2020 at 8:27 | vote | accept | Star | ||
| Jun 8, 2020 at 22:14 | answer | added | whuber♦ | timeline score: 1 | |
| Jun 6, 2020 at 17:52 | comment | added | whuber♦ | @Dilip One need only adopt a relevant definition of "symmetric." For instance, the CDF of a random variable $X$ symmetric about a value $\mu$ satisfies $F_X(\mu+x)=1-F_X(\mu-x)$ for all real numbers $x$ where either of $\mu\pm x$ is a point of continuity. | |
| Jun 6, 2020 at 8:19 | comment | added | Star | I have actually one more doubt related to how to prove the statement: am I using somewhere that the probability mass function or density of $Z$ is symmetric around zero? | |
| Jun 6, 2020 at 8:16 | comment | added | Star | Thanks. I have corrected that part. | |
| Jun 6, 2020 at 8:15 | history | edited | Star | CC BY-SA 4.0 |
added 78 characters in body
|
| Jun 5, 2020 at 19:39 | comment | added | Dilip Sarwate | How can a CDF be symmetric about $0$? CDFs are non decreasing functions. | |
| Jun 5, 2020 at 14:20 | comment | added | whuber♦ | You can do that. However, your approach ignores the essential simplicity of the situation. I think you can obtain better insight by considering the more general situation where $F$ is a mixture of arbitrary distributions and $G$ is also an arbitrary distribution. Then (adopting a suggestive meta-notation), you can write a mixture as $F=\sum_i \pi_iF_i$ and, using $\star$ for the distribution arising from summing the underlying random variables, compute $$F\star G = \left(\sum_i \pi_iF_i\right)\star G = \sum_i\pi_i\left(F_i\star G\right),$$ QED. | |
| Jun 5, 2020 at 14:11 | comment | added | Star | Is it correct they way in which I go from $g$ to $G$? Can I do that? | |
| Jun 5, 2020 at 14:11 | comment | added | Star | This is my attempt following your suggestion. Consider for simplicity the case where $Y$ and $Z$ are discrete. If $Y$ and $Z$ are independent, then the probability mass function of $X\equiv Y+Z$ evaluated at $x$ is $\sum_{l=-\infty}^\infty Pr(Y=l)Pr(Z=x-l)=\sum_{j=1}^J \lambda_j g(x-\mu_j)$, where $g$ is the probability mass function associated with the CDF $G$. In turn, the CDF of $X$ evaluated at $x$ is $\sum_{j=1}^J \lambda_j G(x-\mu_j)$. | |
| Jun 5, 2020 at 14:07 | comment | added | whuber♦ | Simply convolve the distributions. Because convolution is a linear operator, a mixture input becomes a mixture output. This is practically a tautology, but its basic simplicity suggests that a mindless application of the definitions will take you directly to your goal. | |
| Jun 5, 2020 at 13:50 | history | edited | Star | CC BY-SA 4.0 |
added 58 characters in body
|
| Jun 5, 2020 at 13:38 | history | asked | Star | CC BY-SA 4.0 |