# Finding the pmf of a r.v. S ~ Poisson's multinomial distribution. [duplicate]

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Notation

• # categories $$= c$$.
• # trials $$= t$$.
• Side $$i$$ = $$si$$.
• Random vector $$= S = \left[S_1\;S_2\;\ldots\;S_c\right]^T$$.
• # occurrences of $$si$$ vector $$= s = \left[s1\;s2\;\ldots\;sc\right]^T$$.
• Probability of occurrence of $$si$$ vector $$= p = \left[p_{s1}\;p_{s2}\;\ldots\;p_{sc}\right]^T$$.
• $$p_{s1} + p_{s2} + \cdots + p_{sc} = 1$$.
• $$s1 + s2 + \cdots + sc = t$$.
• Pmf of $$S = P\left[S = s\right]$$.

Games

• g1: An ideal 2 sided die is simulated using a standard die by assigning faces w/ dots 1 through 3 & 4 through 6 to $$s1$$ & $$s2$$, respectively. The 2 sided die is tossed 7 times & the # times $$s1$$ & $$s2$$ occur are recorded.
• g2: Same concept as in g1, accept faces w/ dots 1 through 3 have micro holes etched into them s.t. $$p_{s2} = 18/30$$.
• g3: Same concept as in g1, accept w/ the standard die, i.e., $$p_{s1} = p_{s2} = \cdots = p_{s6} = 5/30$$.
• g4: Same concept as in g2, accept w/ the standard die, i.e., $$p_{s4} = p_{s5} = p_{s6} = 6/30$$.
• g5: Same concept as in g2, accept the micro holes are filled w/ $$0.07$$ kg of a material, which evaporates @ $$0.01$$ kg/s upon being sprayed w/ an activator, s.t. $$p_{s2} = 15/30$$ for the 1st toss. Immediately after being sprayed, the outcomes of the tosses are measured every second.
• g6: Same concept as in g5, accept w/ the standard die, i.e., $$p_{s4} = p_{s5} = p_{s6} = 5/30$$ for the 1st toss.

Questions

• q1: Find pmf & evaluate when $$s = \left[2\;5\right]^T$$.
• q2: q1.
• q3: Find pmf & evaluate when $$s = \left[0\;2\;1\;1\;0\;3\right]^T$$.
• q4: q3.
• q5: q1.
• q6: q3.

Answers

• a1: $$S$$ ~ Binomial distribution.
• $$P\left[S = s\right] = t!\prod_{k = 1}^c \frac{p_{sk}^{sk}}{sk!} = 7!\prod_{k = 1}^2 \frac{(15/30)^{sk}}{sk!} = \frac{7!(15/30)^2(15/30)^5}{2!5!}$$
$$\Longrightarrow P\left[S = s\right] = 21/128$$.
• a2: $$S$$ ~ Binomial distribution.
• $$P\left[S = s\right] = t!\prod_{k = 1}^c \frac{p_{sk}^{sk}}{sk!} = 7!\prod_{k = 1}^2 \frac{p_{sk}^{sk}}{sk!} = \frac{7!(12/30)^2(18/30)^5}{2!5!}$$
$$\Longrightarrow P\left[S = s\right] = 20412/78125$$.
• a3: $$S$$ ~ Multinomial distribution.
• $$P\left[S = s\right] = t!\prod_{k = 1}^c \frac{p_{sk}^{sk}}{sk!} = 7!\prod_{k = 1}^6 \frac{(5/30)^{sk}}{sk!} = \frac{7!(5/30)^2(5/30)(5/30)(5/30)^3}{0!2!1!1!0!3!}$$
$$\Longrightarrow P\left[S = s\right] = 35/23328$$.
• a4: $$S$$ ~ Multinomial distribution.
• $$P\left[S = s\right] = t!\prod_{k = 1}^c \frac{p_{sk}^{sk}}{sk!} = 7!\prod_{k = 1}^6 \frac{p_{sk}^{sk}}{sk!} = \frac{7!(4/30)^2(4/30)(6/30)(6/30)^3}{0!2!1!1!0!3!}$$
$$\Longrightarrow P\left[S = s\right] = 224/140625$$.
• a5: $$S$$ ~ Poisson's binomial distribution.
• $$P\left[\left[S_1\;S_2\right]^T = \left[s1\;s2\right]^T\right] = P\left[S_1 = s1, S_2 = s2\right] = P\left[S_1 = s1\right] = P\left[S_2 = s2\right]$$.
• $$p_{s1}$$ & $$p_{s2}$$ are vectors now: $$p_{s1} = \left[p_{s1_1}\;p_{s1_2}\;\ldots\;p_{s1_t}\right]^T, p_{s2} = \left[p_{s2_1}\;p_{s2_2}\;\ldots\;p_{s2_t}\right]^T$$.
• $$P\left[S = s\right] = \frac{1}{t + 1}\sum_{i = 0}^t \left\{\exp\left(\frac{-j2\pi i s}{t + 1}\right) \prod_{k = 1}^t \left\{p_{s2_k}\left(\exp\left(\frac{j2\pi i}{t + 1}\right) - 1\right) + 1\right\}\right\}$$
$$= \frac{1}{8}\sum_{i = 0}^7 \left\{\exp\left(\frac{-j5\pi i}{4}\right) \prod_{k = 1}^7 \left\{\left(\frac{0.5k + 14.5}{30}\right)\left(\exp\left(\frac{j\pi i}{4}\right) - 1\right) + 1\right\}\right\}$$
$$\Longrightarrow P\left[S_2 = 5\right] = 53529/250000$$.
• a6: $$S$$ ~ Poisson's multinomial distribution.
• I understand this enough to write a q about it, but not enough to solve it.
• This q was the only 1 on SE that was relatable but the answer was incorrect.
• PLEASE help me find the exact solution; not an approximation.

## marked as duplicate by Community♦Apr 7 at 2:22

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