Question on HW2.1

[yusunchina]

I was, out of curiosity, trying to derive an exact theoretical expression for the expected value of v_min, and this is what I got (apologies for the poor typesetting):

E( #heads_min ) = sum from i=0 to i=10 ( P(#heads_min = i) * i )
P(#heads_min = i) = 1000 * P( one gets i intersect the others get >= i)
for which *1000 means that out of the 1000 trials, I choose one trial to be the minimum head trial.

P( one gets i intersect the others get >= i)
= P( one gets i ) * P(others get >= i)
= p(i, 10, 0.5) * [1 - F(i-1, 10, 0.5)] ^ 999
for which p is the probability mass function and F is the cumulative mass function, with parameters (number successes, total number, prob. of success). I used i-1 for F since >= i includes i itself.

and this expression yielded the result E( #heads_min ) = 3.6814, and expected v_min is thus 0.36814, which is incorrect.
Would anyone please point to me where my mistake is? I would really really appreciate it!

Here is the R code I used for the calculation:

phmin = function(i) {
u = 0.5 # probability of head
N = 10 # number of flips per trial
m = 1000 # number of trials
one_min = dbinom(i, N, u)
others_more = pbinom(i-1, N, u, lower.tail = FALSE) ^ (m-1)
return(1000 * one_min * others_more)
}

ivals = (0:10)
expect_hmin = sum(phmin(ivals) %*% ivals)
expect_hmin / 10

[yaser]

Originally Posted by yusunchina

P(#heads_min = i) = 1000 * P( one gets i intersect the others get >= i)

This formula assumes that the events "one gets i intersect the others get >= i" (with the "one" fixed at one coin at a time) are disjoint, otherwise this would be an over-count. These events are in fact overlapping. For instance, the case where all the coins get exactly i (which is admittedly an extremely unlikely event but chosen here as a simple illustration) would be counted in all 1000 cases, although it happens only "once."

[yusunchina]

Thank you so much, this clears it up exactly and I changed it and got the correct answer! I'll keep working.