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 solve this problem, we need to use the properties of a Poisson distribution and make use of the given information.
Let's start with the given equation: 3p(x=2) = 2p(x=1).
1. To find p(x=0), we can use the fact that the sum of probabilities for all possible values of x is equal to 1. So, we need to find the value of p(x=0) such that it satisfies this equation.
Let's assume p(x=0) = a. Now we have:
3p(x=2) + 2p(x=1) + a = 1
We already know that 3p (x=2) = 2p (x=1). Substituting this value into the equation, we get:
2(2p(x=1)) + 2p(x=1) + a = 1
4p(x=1) + 2p(x=1) + a = 1
6p(x=1) + a = 1
Now, we know that the sum of probabilities, p(x=0) + p(x=1) + p(x=2), is equal to 1. So, we can write p(x=1) + p(x=2) = 1 - p(x=0).
Substituting this into the previous equation, we get:
6(1 - p(x=0)) + a = 1
6 - 6p(x=0) + a = 1
6p(x=0) + a = 5
Now, we have two equations:
6p(x=1) + a = 1 -- (Equation 1)
6p(x=0) + a = 5 -- (Equation 2)
To find p(x=0), we can subtract Equation 2 from Equation 1:
6p(x=1) - 6p(x=0) = 1 - 5
6(p(x=1) - p(x=0)) = -4
We know that p(x=1) - p(x=0) = p(x=1) - 1 = -4/6 = -2/3.
Solving this equation, we get p(x=1) = 1 - 2/3 = 1/3.
Now, substituting p(x=1) = 1/3 in Equation 1, we can find a:
6(1/3) + a = 1
2 + a = 1
a = -1
Therefore, p(x=0) = a = -1.
2. To find p(x=3), we can use the given value e^(-4/3) = 0.264. In a Poisson distribution, the probability mass function is given by p(x) = (e^(-λ) * λ^x) / x!, where λ is the average rate of the Poisson distribution.
Comparing this formula with the given value, we can equate the two and solve for x:
(e^(-λ) * λ^x) / x! = e^(-4/3)
Using the given value e^(-4/3) = 0.264, we have:
(e^(-λ) * λ^x) / x! = 0.264
Since we are looking for p(x=3), we set x = 3. So we have:
(e^(-λ) * λ^3) / 3! = 0.264
Simplifying this equation:
(e^(-λ) * λ^3) / 6 = 0.264
Multiplying both sides by 6:
e^(-λ) * λ^3 = 0.264 * 6
Using the given value e^(-4/3) = 0.264, we can substitute:
(0.264 * e^(-4/3) * λ^3) = 0.264 * 6
Cancelling out the common factor of 0.264:
e^(-4/3) * λ^3 = 6
Taking the natural logarithm of both sides:
ln(e^(-4/3) * λ^3) = ln(6)
Using the properties of logarithms:
(-4/3) + 3ln(λ) = ln(6)
Simplifying this equation:
3ln(λ) = ln(6) + (4/3)
Dividing both sides by 3:
ln(λ) = (ln(6) + (4/3))/3
Taking the exponential of both sides:
λ = e^((ln(6) + (4/3))/3)
Now, substitute the given value of e^(-4/3) = 0.264 into the equation to find λ:
0.264 = e^((ln(6) + (4/3))/3)
Taking the natural logarithm of both sides and solving for λ:
ln(0.264) = (ln(6) + (4/3))/3
Simplifying the equation:
ln(0.264) = ln(6) + (4/3)
Subtracting ln(6) from both sides:
ln(0.264) - ln(6) = 4/3
Taking the exponent of both sides to solve for λ:
e^(ln(0.264) - ln(6)) = e^(4/3)
Simplifying the left side using the properties of logarithms:
0.264/6 = e^(4/3)
Multiplying both sides by 6:
0.264 = 6 * e^(4/3)
Dividing both sides by 6:
0.264/6 = e^(4/3)
Taking the cube root of both sides:
(0.264/6)^(1/3) = e^(4/9)
Using a calculator, we can evaluate the left side:
(0.264/6)^(1/3) = 0.56757
Therefore, λ = e^(4/9) ≈ 0.56757.
Now that we have found λ, we can calculate p(x=3) using the formula for a Poisson distribution:
p(x=3) = (e^(-λ) * λ^3) / 3!
Substituting the value of λ:
p(x=3) = (e^(-0.56757) * (0.56757)^3) / 3!
Using a calculator, we can evaluate this expression:
p(x=3) ≈ 0.1387.
Therefore, p(x=3) ≈ 0.1387.
To summarize:
1. p(x=0) = -1
2. p(x=3) ≈ 0.1387.