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 ? but slightly different means $?_A$ and $?_B$, where $?_A>?_B$. Let $\delta_\mu = ?_A-?_B$.

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: $n = 6, q = [1,2,3,4,5], ?_A=1001$ and $?_B=1000$, and $?=10$.

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

  • Yes, that is what n represents.

  • Thanks

Answer

Hi,

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

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

  • Hey, no problem. Glad I could help

The answer is accepted.