# distribution of positive cases in random sample

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?

• Have you already determined what the probability is that one person tests positive (situation where m=1)? Nov 10, 2020 at 8:52

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.