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I need help with a question I am trying to work out:

Individuals have 5 choices:

  • Choice 1: utility1 = alpha * R1 + beta * C1 + random_shock_1(mu=0,sig=1)
  • Choice 2: utility2 = alpha * R2 + beta * C2 + random_shock_2(mu=0,sig=1)
  • Choice 3: utility3 = alpha * R3 + beta * C3 + random_shock_3(mu=0,sig=1)
  • Choice 4: utility4 = alpha * R4 + random_shock_4(mu=0,sig=1)
  • Choice 5: utility5 = beta * C4 + random_shock_5(mu=0,sig=1)

There are some mass M of individuals who will choose the option that has the highest utility. However, they each have different alphas and betas, i.e. preference for consuming R or C, where R1, R2, etc are different values or R and C1, C2 etc are different values of C aka constants. However the distribution of alpha and beta is known (i.e. we know the mean and sd of both).

Is it possible to calculate the expected proportion of individuals that will choose each of the choices? and if so how?

EDIT: Update below

Random normal variables: (alpha and beta) follow normal distributions. e.g. $\alpha \sim \mathcal{N}(\mu_{\alpha},\sigma_{\alpha}^{2})$ and $\beta \sim \mathcal{N}(\mu_{\beta},\sigma_{\beta}^{2})$. Both alpha, beta and the shocks are independent of each other.

Constants: R1, R2, C1 etc are constants.

I am ending up with 5 different distributions (e.g.utility1, utility2, ...) that are follow random normal distributions.

My question is close to What is $P(X_1>X_2 , X_1>X_3,... , X_1>X_n)$? and I have tried following this approach, but I am trying to avoid numerical integration by simulation.

I am coding this up in fortran and have a function to calculate the PDF and CDF of the variables, but unsure if I am still required to integrate over them from $-\infty$ to $\infty$.

Jeremy Miles
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Laurence_jj
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  • What would be the basis of the calculation? What data and/or assumptions would you be using? – whuber Jan 19 '24 at 19:37
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    So I am trying to work out the expected number of individuals who would choose one option over another as simulating a large number of individuals in my situation is highly costly. In essence I am therefore trying to work out the proportion who would choose say choice 1, I am thinking the answer may lie in -> https://stats.stackexchange.com/questions/44139/what-is-px-1x-2-x-1x-3-x-1x-n?lq=1&fbclid=IwAR2vBHh2Q6Vt1STu2g8ZyAC-hKNZHGLydd6ldQsm_HpXvhrCY-nSZLlPCJw – Laurence_jj Jan 19 '24 at 20:54
  • Thanks @whuber for your comments I have added context to the answer to help clarify. I saw your response to the other question which I found most helpful, and I was wondering if you could help me in my simplified case. All random variables are independent – Laurence_jj Jan 22 '24 at 20:33

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