Poisson Distribution is a mathematical concept that helps to calculate the probability of an event occurring in a given time frame. It is used when the number of occurrences of an event is rare but can happen at any time. The formula for Poisson Distribution is:

P (x; λ) = (e^-λ) (λ^x) / x!

Where x is the number of occurrences, λ is the mean number of occurrences, and e is the base of the natural logarithm.



To calculate the probability of a Poisson variety lying between -2.3 to 3.68, we need to follow these steps:

1. Find the mean number of occurrences (λ)

The standard deviation of a Poisson variety is equal to the square root of the mean. Therefore, the mean number of occurrences can be calculated as:

SD = sqrt(λ)
1.732 = sqrt(λ)
λ = (1.732)^2
λ = 3

2. Calculate the probability of x for each value within the range

We need to calculate the probability of x for each value within the range using the Poisson Distribution formula:

P (x; λ) = (e^-λ) (λ^x) / x!

For x = -2.3:
P (-2.3; 3) = (e^-3) (3^-2.3) / (-2.3)!
P (-2.3; 3) = 0.007

For x = -1.3:
P (-1.3; 3) = (e^-3) (3^-1.3) / (-1.3)!
P (-1.3; 3) = 0.023

For x = -0.3:
P (-0.3; 3) = (e^-3) (3^-0.3) / (-0.3)!
P (-0.3; 3) = 0.086

For x = 0.68:
P (0.68; 3) = (e^-3) (3^0.68) / 0.68!
P (0.68; 3) = 0.302

For x = 1.68:
P (1.68; 3) = (e^-3) (3^1.68) / 1.68!
P (1.68; 3) = 0.321

For x = 2.68:
P (2.68; 3) = (e^-3) (3^2.68) / 2.68!
P (2.68; 3) = 0.201

For x = 3.68:
P (3.68; 3) = (e^-3) (3^3.68) / 3.68!
P (3.68; 3) = 0.086

3. Sum the probabilities of all the values within the range

The probability of the Poisson variety lying between -2.3 to 3.68 is the sum of all the probabilities calculated in step 2:

P (-2.3 ≤ x ≤ 3

0.65