distribution of positive cases in random sample

Question

If I know the true distribution of the number of people who are in each category in a population, how can I determine the probability that test on a random sample will have a certain range of positive cases. Specifically suppose there are $n$ people in the population divided among three categories [$x_1$,$x_2$,$x_3$] so that $x_1+x_2+x_3=n$, and I have test where the probability a person in category $x_i$ tests positive is $p_i$. If I take a random sample of size $m$, how can I determine the probability that the number of positive tests will be in some range?

Answer 1

Let's denote by $T$ the number of positive tests in your sample, and by $T_1, T_2$ and $T_3$ the number of positive tests in categories 1, 2 and 3. Then $T = T_1 + T_2 + T_3$.

First, let's consider that $x_1, x_2, x_3$ are fixed. Then $T_1, T_2$ and $T_3$ are independent binomial random variables : $$\begin{array}{ccc} T_1 & \sim & \mathcal{B}(x_1, p_1)\\ T_2 & \sim & \mathcal{B}(x_2, p_2) \\ T_3 & \sim & \mathcal{B}(x_3, p_3) \end{array}$$ So $T$, is... the sum of three independent binomial random variables (not much more we can say if you don't have $p_1 = p_2 = p_3$). I think you can easily get the probability distribution of $T$ numerically, but not in closed form. However, if $x_1\times p_1$, $x_2 \times p_2$ and $x_3 \times p_3$ are big enough, you can approximate these binomial distributions by normal distributions : $$\begin{array}{ccc} T_1 & \sim & \mathcal{N}\left(x_1 p_1, x_1 p_1 (1-p_1)\right)\\ T_2 & \sim & \mathcal{N}\left(x_2 p_2, x_2 p_2 (1 - p_2)\right) \\ T_3 & \sim & \mathcal{N}\left(x_3 p_3, x_3 p_3 (1 - p_3)\right) \end{array}$$ such that $T\sim\mathcal{N}\left(x_1 p_1 + x_2 p_2 + x_3 p_3, x_1 p_1 (1 - p_1)+ x_2 p_2 (1 - p_2) + x_3 p_3 (1 - p_3)\right)$.

Now, let's take into account the fact that $x_1, x_2$ and $x_3$ are not fixed but realizations of random variables $X_1, X_2, X_3$ whose joint distribution is a multinomial (or polytomous) with parameters $n, p_1, p_2, p_3$. Then $$P[T = t] = \mathbb{E}_{X_1, X_2, X_3}[P(T = t \mid X_1, X_2, X_3)]$$ where the conditionnal probability is given either by the sum of binomials if you don't want to use the normal approximations, either by the normal density of $T$ above if you can use the approximation.

This expectation can be computed numerically by simulations or by numerical integration methods. Unfortunately, I don't think it can easily be computed analytically.

Hope this is helpful.