Question

Assume a Poisson distribution. Find the following probabilities. a. Let λ=2.0, find P(x≥2). b. Let λ=0.5, find P(X≤1). c. Let λ=4.0, find P(X≤3). d. Let λ=42, find P(X≥1). e. Let λ=5.6, find P(X≤2).

Answer 1

a.
`\( P(X \geq 2) \)` when
`\( \lambda = 2.0 \)` is approximately 0.7358.

b.
`\( P(X \leq 1) \)` when
`\( \lambda = 0.5 \)` is approximately 0.6065.

c.
`\( P(X \leq 3) \)` when
`\( \lambda = 4.0 \)` is approximately 0.6288.

d.
`\( P(X \geq 1) \)` when
`\( \lambda = 42 \)` is approximately 1.0.

e.
`\( P(X \leq 2) \)` when
`\( \lambda = 5.6 \)` is approximately 0.4912.

Step-by-step explanation:

In a Poisson distribution, the probability mass function for a given value x is given by P X = x =
`(e^(-\lambda) \lambda^x)/(x!) \)`, where
`\( \lambda \)` is the average rate of events per unit time or space.

• For
  `\( P(X \geq 2) \)` when
  `\( \lambda = 2.0 \)`, we sum the probabilities for
  `\( x = 2, 3, 4, ... \)`. Calculating this using the formula, we find
  `\( P(X \geq 2) \)` to be approximately 0.7358.

• For
  `\( P(X \leq 1) \)` when
  `\( \lambda = 0.5 \)`, we sum the probabilities for
  `\( x = 0, 1 \)`. Using the formula,
  `\( P(X \leq 1) \)` is approximately 0.6065.

• 
  `\( P(X \leq 3) \)` when
  `\( \lambda = 4.0 \)` involves summing the probabilities for
  `\( x = 0, 1, 2, 3 \)`. The calculation results in approximately 0.6288.

• 
  `\( P(X \geq 1) \)` when
  `\( \lambda = 42 \)` is straightforward because the Poisson distribution includes all non-negative integers. Therefore,
  `\( P(X \geq 1) \)` is 1.0.

• For
  `\( P(X \leq 2) \)` when
  `\( \lambda = 5.6 \)`, we sum the probabilities for
  `\( x = 0, 1, 2 \)`. The calculation gives us approximately 0.4912.