In a Poisson distribution P(X=x) for x=0 is 10%. Find the (Log 102.3026.) mean and P(X<3),
To find the mean of a Poisson distribution, we can use the formula:
Mean = λ
where λ is the average rate of the events occurring. In this case, we are given that P(X=0) = 10%.
P(X=0) = (λ^0 * e^-λ) / 0! = 0.10
Let's solve for λ:
(λ^0 * e^-λ) / 0! = 0.10
e^-λ = 0.10
-λ = ln(0.10)
λ = -ln(0.10)
λ ≈ 2.3026
Therefore, the mean of the Poisson distribution is 2.3026.
To find P(X<3), we can use the Poisson distribution formula:
P(X<3) = P(X=0) + P(X=1) + P(X=2)
We can use the formula for the Poisson distribution:
P(X=x) = (λ^x * e^-λ) / x!
P(X=1) = (2.3026^1 * e^-2.3026) / 1! ≈ 0.2111
P(X=2) = (2.3026^2 * e^-2.3026) / 2! ≈ 0.2420
Now we can calculate:
P(X<3) = P(X=0) + P(X=1) + P(X=2) ≈ 0.10 + 0.2111 + 0.2420 ≈ 0.5531
Therefore, P(X<3) ≈ 0.5531.