Probability maximum value of samples from different distributions

Assume you are sampling values at random from two normal distributions, Distribution A and Distribution B. Both distributions have the same standard deviation σ but slightly different means $\mu_A$ and $\mu_B$, where $\mu_A>\mu_B$. Let $\delta_\mu = \mu_A-\mu_B$.

You're drawing a total of n samples from these two distribution: q samples from A, and n-q from B. What is the probability that the largest value among the n samples came from Distribution A? Please explain your steps. Assume n is finite and relatively small (<100). Verify your results by calculating the probability that the largest sample comes from A for the following parameters: $n = 6, q = [1,2,3,4,5], \mu_A=1001$ and $\mu_B=1000$, and $\sigma=10$.

Statistics Probability Integrals
Cpcpcp Cpcpcp
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  • Jbarth Jbarth
    +1

    Hey there- I'm working on the answer, but wanted to clarify that n represents the total sample size between the two distributions?

  • Cpcpcp Cpcpcp
    +1

    Yes, that is what n represents.

  • Jbarth Jbarth
    +1

    Thanks

Answer

Hi,

Attached is the solutions, with R code in the Appendix. Thanks!

Jbarth Jbarth
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Matchmaticians Probability maximum value of samples from different distributions File #1 File #1 (pdf)
  • Cpcpcp Cpcpcp
    0

    Many thanks for the great and detailed answer. It's very much appreciated. This was a bit too difficult for me to solve.

  • Jbarth Jbarth
    0

    Hey, no problem. Glad I could help

The answer is accepted.
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