Timeline for Expected value of a ratio of squared normal and linear combination of squared normal
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 13, 2021 at 13:36 | comment | added | Iosif Pinelis | @Edward : Actually, for $M=3$ the integral can be expressed in terms of elliptic functions -- see the other answer of mine, updated. | |
| Sep 12, 2021 at 21:38 | comment | added | Iosif Pinelis | @Edward : I think the method of my other answer would work better in the general case. However, even for $M=3$, the corresponding integral, $\int_0^\infty du\, (1+2u)^{-3/2} (1+2cu)^{-1/2}(1+2\tilde cu)^{-1/2}$, apparently cannot be expressed in closed form; Mathematica cannot do anything with it. Closed form expressions are rare. | |
| Sep 12, 2021 at 21:02 | comment | added | Edward | Thank you. Do you have maybe an idea how to calculate the expectation for the more general case $\frac{x_1^2}{\sum_{i=1}^{M}c_i x_i^2}$ where $c_i>0$ and $x_i\sim N(0,1)$ ? (or at least for $M=3$) | |
| Sep 12, 2021 at 13:12 | vote | accept | Edward | ||
| Sep 12, 2021 at 12:29 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 206 characters in body
|
| Sep 12, 2021 at 2:39 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
deleted 8 characters in body
|
| Sep 12, 2021 at 0:57 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |