# Find PMF, Expectation and Variance.

Question

Lets pick $m$ coupons such that each coupon comes with a value from {1,...,n} uniformly at random. Let $X=(\displaystyle\sum_{i=1}^{m} X_i)$ mod $k$, where $X_i$ is the random variable that denotes the outcome of the ${i}^{th}$ coupon and $k$ is a positive integer that divides n. Find the PMF (probability mass function) of the random variable $X$ and the compute $E[X]$ and $V[X]$. Note that $V[X]$ denotes the variance of the random variable $X$.

I don't have answer for this question, I just want to verify my answer and correctness of understanding.

I defined $X$ as values containing $0$ to $k-1$, after that I applied this steps

Calculation of the PMF of a Random Variable $X$ For each possible value $x$ of $X$:

1. Collect all the possible outcomes that give rise to the event ${X = x}$.
2. Add their probabilities to obtain $p_X(x)$.

From that I get my PMF as $m/k$.

• $p_X^{\,}(x) = \dfrac{m}{k}$ is unlikely to be correct, as you should have $\displaystyle \sum_{x=0}^{k-1} p_X^{\,}(x) = 1$ – Henry Aug 27 '17 at 15:40
• then addition of $1/k$ from $0$ to $1/k$, according to second point? – user146551 Aug 27 '17 at 15:54

Comment: I don't much like your notation, but your description seems substantially correct.

Let's denote the totals as $T$ and then as $T_k$ upon taking the 'mod'. You have to understand that the relative frequencies $r_t \approx 1/k,$ for $t = 0, 1, \dots k,$ only estimate the $p_{T_k}(t).$ To me it seems plausible, but not obvious, that the distribution of $T_k$ is discrete uniform. You might want to give an argument for that.

However, a simulation of 100,000 totals from sampling $m = 20$ coupons numbered $1$ through $n=10$ and with modulus $k = 3$ does seem to give the discrete uniform distribution on $\{0,1,2\}.$ (Certainly, a simulation with selected parameters is not a general proof.)

In case it is of interest, here is my simulation in R statistical software.

B = 10^5;  m = 20;  n = 10;  k = 3
x = sample(1:n, B*m, rep=T)
MAT = matrix(x, nrow=B)       # B x m matrix: each row a sample of m
t = rowSums(MAT)
t.k = t %% k                  # modular arithmetic
table(t.k)/B
## t.k
##       0       1       2
## 0.33344 0.33284 0.33372