Question

Let X₁ and X₁ be independent binomial random variables with Xi having parameters (nᵢ, pᵢ), i = 1,2. Find

a. P(X₁X₂ = 0);
b. P(X₁+X₂ = 1);
c. P(X₁+X₂ = 2).

Answer 1

To find the probabilities of different outcomes based on the sum and product of two independent binomial random variables.

Step-by-step explanation:

Let X₁ ~ B(n₁, p₁) and X₂ ~ B(n₂, p₂) be independent binomial random variables.

a. To find P(X₁X₂ = 0), we need to find the probability that both X₁ and X₂ are equal to 0. Since X₁ and X₂ are independent, this probability is equal to P(X₁ = 0) * P(X₂ = 0). Using the binomial probability formula, we can calculate these probabilities and multiply them to find the final probability.

b. To find P(X₁ + X₂ = 1), we need to find the probability that the sum of X₁ and X₂ is equal to 1. We can calculate this probability using the binomial probability formula.

c. To find P(X₁ + X₂ = 2), we need to find the probability that the sum of X₁ and X₂ is equal to 2. Again, we can use the binomial probability formula to calculate this probability.