If x is a Poisson variate such that
3p (x=2) = 2p (x=1).
Find
1. p (x=0)
2. p (x-3)
Given e^(-4/3)=0.264
To find the answers to the given questions, we need to use the properties of the Poisson distribution and solve the equations provided. Let's break down the steps:
1. To find p(x=0), we can use the fact that the sum of probabilities for all possible values of x in a Poisson distribution equals 1.
The probability mass function (PMF) of a Poisson distribution is given by:
P(x) = (e^(-λ) * λ^x) / x!
In this case, we are given that:
3P(x=2) = 2P(x=1)
Let's write the equations:
3 * (e^(-λ) * λ^2) / 2! = 2 * (e^(-λ) * λ^1) / 1!
Simplifying the equation, we can cancel out the common terms:
3 * (λ^2) / 2! = 2 * (λ^1) / 1!
λ^2 / 2 = 2λ
Now, rearrange the equation:
λ^2 = 4λ
This is a quadratic equation. To solve it, we bring all the terms to one side:
λ^2 - 4λ = 0
Factoring out λ, we get:
λ(λ - 4) = 0
From this equation, we have two possible values: λ = 0 and λ = 4.
Since λ represents the average rate of events in a given time period, it cannot be zero (as this would mean no events occur) but it can be 4. Therefore, we choose λ = 4.
Using the PMF formula, let's find p(x=0) using λ = 4:
P(x=0) = (e^(-4) * 4^0) / 0!
Since 0! = 1, we have:
P(x=0) = e^(-4)
Given that e^(-4/3) = 0.264, we can conclude that p(x=0) = 0.264.
2. To find p(x=3), we can use the PMF formula once again. Using λ = 4:
P(x=3) = (e^(-4) * 4^3) / 3!
Since 3! = 3 * 2 * 1 = 6, we have:
P(x=3) = (e^(-4) * 64) / 6
Now, we can calculate p(x=3) using the given value of e^(-4/3):
0.264 = (e^(-4) * 64) / 6
Multiply both sides of the equation by 6 to isolate (e^(-4)):
1.584 = e^(-4) * 64
Divide by 64 to solve for e^(-4):
0.02475 = e^(-4)
Therefore, p(x=3) = 0.02475.