A3. A random variable X has a Poisson distribution such that P(X= 1|X≤1) = 0.75,find the V ar(X

The problem is asking for the variance of a Poisson distributed random variable X, given that the conditional probability P(X=1|X≤1) = 0.75.

Step 1: Understand the Poisson Distribution The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space. The variance of a Poisson distribution is equal to its mean (λ).

Step 2: Understand the given condition The condition given is P(X=1|X≤1) = 0.75. This is the probability that X equals 1 given that X is less than or equal to 1.

Step 3: Use the properties of the Poisson distribution For a Poisson distribution, P(X=k) = λ^k * e^-λ / k!. Given that X is less than or equal to 1, there are only two possible values for X: 0 and 1. Therefore, P(X=1|X≤1) is actually P(X=1) / (P(X=0) + P(X=1)).

Step 4: Solve for λ We know that P(X=1|X≤1) = 0.75, so we can set up the equation 0.75 = P(X=1) / (P(X=0) + P(X=1)) and solve for λ.

P(X=1) = λ^1 * e^-λ / 1! = λ * e^-λ P(X=0) = λ^0 * e^-λ / 0! = e^-λ

Substituting these into the equation gives us 0.75 = λ * e^-λ / (e^-λ + λ * e^-λ). Solving this equation for λ will give us the mean of the distribution.

Step 5: Find the variance Since the variance of a Poisson distribution is equal to its mean, the variance Var(X) is also equal to λ.

Please note that the equation in step 4 is transcendental and cannot be solved analytically for λ. It requires numerical methods to solve.