Finding the expectation of a categorical variable times a random amount

External question — Cross Validated Stack Exchange Author: Vefeagins Original post: https://stats.stackexchange.com/questions/652555 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. Say we have $J$ trading cards and each have a dollar value of $u_{j}$ and I am allowed to make 1 draw. $$ u_{j} \sim N(\mu,1) $$ Where the value of each trading card is independent and identically distributed. The probability of drawing a trading card is a function of $u_{j}$ . We could specify this as a multinominal regression for the $J$ cards. $$ \text{ln Pr}(C_i = j) = u_{j} - \text{ln} Z_i $$ $$ Z_i = \sum_{j}^J e^{u_{j}} $$ $Z_i$ is a normalizing value to ensure the probabilities add up to one. Where $C_i$ is a categorical distribution where each probablity $p_j$ is given with the above formula. This is also the same as multinominal distribution with $n= 1$ . $$ C_i \sim Categorical(p_1^{[C_i = 1]} \cdots p_J^{[C_i = J]}) $$ The random value of the card would be: $$ u_{1C_i} = \sum_j^J [C_i = j] * u_{j} $$ If I wanted to know the expected value of dollar amount from drawing one card. How would I set that up? $$ {\bf E} (u_{C_i}) = \int u_{C_i} f_{pdf}(u_{C_i}) $$ I am interested in setup what the integral or sum would look like? I understand that it will not simplify into a nice, closed form answer. I was struggling with how to handle that the probability of $\text{Pr}(C_i = j)$ depends on the set $U = \{u_1, \cdots u_J\}$ and not just a particular $u_{j}$ . The motivating issue for this problem has to do with a complicated multinominal regression where there are random beta coefficients.

External answer — Cross Validated Stack Exchange Author: JimB Original post: https://stats.stackexchange.com/a/652602 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. If you are just interested in setting up the integrals for the expectation, then maybe the following does that: $$\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\frac{\sum _{i=1}^J e^{u_i} u_i}{\sum _{i=1}^J e^{u_i}} \times \prod _{i=1}^J \frac{e^{-\frac{1}{2} (u_i-\mu)^2}}{\sqrt{2 \pi }} du_1 \cdots du_J$$ $$=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\frac{\sum _{i=1}^J e^{v_i+\mu} (v_i+\mu)}{\sum _{i=1}^J e^{v_i+\mu}} \times \prod _{i=1}^J \frac{e^{-\frac{1}{2} v_i^2}}{\sqrt{2 \pi }} dv_1 \cdots dv_J$$ $$=\mu+\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\frac{\sum _{i=1}^J e^{v_i} v_i}{\sum _{i=1}^J e^{v_i}} \times \prod _{i=1}^J \frac{e^{-\frac{1}{2} v_i^2}}{\sqrt{2 \pi }} dv_1 \cdots dv_J$$ Numerical integration will work fine for small values of $J$ but you'll likely need to perform simulations for values of $J$ greater than 4.