Question

Assume a Poisson distribution. Find the following a. Let λ=7.0, find P(X≥3). b. Let λ=0.4, find P(X≤1). c. Let λ=2.0, find P(X≤2). d. Let λ=4.1, find P(X≥2). e. Let λ=5.1, find P(X≤2).

Answer 1

a. P(X≥3) ≈ 0.6161 b. P(X≤1) ≈ 0.6703 c. P(X≤2) ≈ 0.8571 d. P(X≥2) ≈ 0.6414

e. P(X≤2) ≈ 0.1847

Step-by-step explanation:

In a Poisson distribution, λ represents the average rate of occurrence within a specific interval. To calculate probabilities, the formula for the Poisson distribution is P(X=k) = (e^(-λ) * λ^k) / k!, where X is the random variable, λ is the average rate, and k is the number of occurrences.

a. For P(X≥3) with λ=7.0: Using the complement rule (P(X≥3) = 1 - P(X<3)), calculate P(X<3) and then subtract from 1.

b. For P(X≤1) with λ=0.4: Directly compute P(X=0) and P(X=1), then add these probabilities together.

c. For P(X≤2) with λ=2.0: Compute the cumulative probability of P(X=0) + P(X=1) + P(X=2).

d. For P(X≥2) with λ=4.1: Use the complement rule, calculate P(X<2) and then subtract from 1 to find P(X≥2).

e. For P(X≤2) with λ=5.1: Compute the cumulative probability of P(X=0) + P(X=1) + P(X=2). Calculate each using the Poisson distribution formula and sum the individual probabilities.