Question

Calculations for a Poisson Distribution:

If X ~ P(2.1), calculate:
a) P(X=0)
b) P(X ≤ 2)
c) P(X ≥ 5)
d) P(X=1 | X ≤ 2)

Answer 1

For a Poisson distribution with a mean of 2.1, the probabilities are calculated as follows: (a) P(X=0) = 0.1221, (b) P(X ≤ 2) = 0.6451, (c) P(X ≥ 5) = 0.3549, and (d) P(X=1 | X ≤ 2) = 0.3965.

Step-by-step explanation:

In a Poisson distribution, the mean (denoted by μ) is given as 2.1.

a) To calculate P(X=0), we can use the Poisson probability mass function (pmf) formula: P(X=x) = (e^(-μ) * μ^x) / x!. Substituting the values, we get P(X=0) = (e^(-2.1) * 2.1^0) / 0! = 0.1221.

b) To calculate P(X ≤ 2), we can use the Poisson cumulative distribution function (cdf) formula: P(X ≤ x) = sum(P(X=i) for i=0 to x). Substituting x=2, we get P(X ≤ 2) = P(X=0) + P(X=1) + P(X=2) = 0.1221 + 0.2554 + 0.2676 = 0.6451.

c) To calculate P(X ≥ 5), we can use the complement rule: P(X ≥ 5) = 1 - P(X ≤ 4). Using the cdf formula, we find P(X ≤ 4) = 0.6451, so P(X ≥ 5) = 1 - 0.6451 = 0.3549.

d) To calculate P(X=1 | X ≤ 2), we can use the conditional probability formula: P(A | B) = P(A ∩ B) / P(B). Here, A is the event X=1 and B is the event X ≤ 2. We already calculated P(X=1) as 0.2554 and P(X ≤ 2) as 0.6451, so P(X=1 | X ≤ 2) = P(X=1 ∩ X ≤ 2) / P(X ≤ 2) = P(X=1) / P(X ≤ 2) = 0.2554 / 0.6451 = 0.3965.