[Math] Product of the three numbers obtained by rolling three dice. Expectation and Variance

expectationprobabilityvariance

What is the expected value and variance of X, the product of the three numbers obtained by rolling three fair die?

I tried solving this problem by dividing the numbers $\{1,…,216\}$ into primes and composites, but realized there were too many. The hint that was given to me was to use indicators. I can't figure out how.

Best Answer

It seems we're down to the 'good practice' of brute force, in which (for me anyhow) bookkeeping errors might creep in. So I'll provide a simulation which may be accurate enough to help catch mistakes. About three digits of accuracy for $E(X)$ and $SD(X).$

set.seed(421);  m=10^7
x = replicate( m, prod(sample(1:6, 3, repl=T))  )
mean(x);  sd(x)
## 42.86178  # aprx E(X) = (7/2)^3 = 42.875
## 40.62558  # aprx SD(X) = 40.62621
sqrt((91/6)^3 - (7/2)^6)
## 40.62621

Numerators of 216 (2nd row):

 round(table(x)*216/m)
 x
 1   2   3   4   5   6   8   9  10  12  15  16  18  20  24  25  27  30  32  36 
 1   3   3   6   3   9   7   3   6  15   6   6   9   9  15   3   1  12   3  12 

40  45  48  50  54  60  64  72  75  80  90  96 100 108 120 125 144 150 180 216 
 6   3   9   3   3  12   1   9   3   3   6   3   3   3   6   1   3   3   3   1
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