# Calculate number of defects(PMF of sum of multinomial distributions)

Suppose you have two factories, A and B which produce two items at the same time with a probability of defect from A, $$p_A$$ and probability of defect from B, $$p_B$$

Suppose a total of $$N_T$$ items are made so $$\frac{N_T}{2}$$ from each. How would you compute the probability that $$N_d$$ items are defective?

It looks like you can consider this to be a sum of multinomial distributions as supposing $$X_A$$ represents the number of defects out of $$\frac{N_T}{2}$$ from factory A and $$X_B$$ represents the number of defects out of $$\frac{N_T}{2}$$ from factory B both are distributed according to multinomial distribution and we are interested in $$P(X_A + X_B = N_d)$$

I don't see any closed form solution to computing this probability without summing the probability of every combination which sums to $$N_d$$

• Given $N_T$ is large you can use normal approximation for binomial – kludg Dec 5 '19 at 7:03
• If $p_A=p_B, X_A+X_B$ is also a binomial distribution with parameters $N_T, p_A$. – Shubham Johri Dec 5 '19 at 7:20
• The result of the summation can be expressed using the hypergeometric function; I doubt that you'll find a more closed form for it than that. – joriki Dec 5 '19 at 23:01