Question

Let the independent random variables \(X_{1}\) and \(X_{2}\) have binomial distribution with parameters \(n_{1}=3, p=\frac{2}{3}\) and \(n_{2}=4, p=\frac{1}{2}\), respectively. Compute \(P\left(X_{1}=X_{2}\right)\) Hint: List the four mutually exclusive ways that \(X_{1}=X_{2}\) and compute the probability of each.

Short answer

To get the final probability that \(X_{1} = X_{2}\), sum up the probabilities for each of the four exclusive scenarios where both \(X_{1}\) and \(X_{2}\) have the same value.

Step-by-step solution

Identify the mutually exclusive ways

Given that \(X_{1}\) has parameters \(n_{1} = 3\) and \(X_{2}\) has parameters \(n_{2} = 4\), the possible mutually exclusive ways that \(X_{1} = X_{2}\) are when both are 0, 1, 2, 3. Only compute till 3 since \(n_{1} = 3\) is the maximum for \(X_{1}\).

Use the binomial distribution formula

Use the binomial formula to compute the probability for each case. The formula is \[P(k) = \binom{n}{k}p^k(1-p)^{n-k}\] where \(n\) is the number of trials, \(p\) is the success probability, and \(k\) is the specific number of successes in \(n\) trials.

Compute the probabilities

For 0, compute \[P(X_{1} = 0)P(X_{2} = 0)\], for 1, compute \[P(X_{1} = 1)P(X_{2} = 1)\], for 2, compute \[P(X_{1} = 2)P(X_{2} = 2)\], and for 3, compute \[P(X_{1} = 3)P(X_{2} = 3)\]. Substitute the parameters into the formula to get these probabilities.

Sum the probabilities

Summing up these probabilities will give the final probability that \(X_{1} = X_{2}\). This can be represented as \[P(X_{1} = X_{2}) = \sum _{i=0} ^3 P(X_{1} = i)P(X_{2} = i)\]