Assuming lambda=0.02 and t=50, compute P(2)
To compute P(2) given lambda=0.02 and t=50, we can use the Poisson distribution formula:
P(k) = (lambda^k * e^(-lambda)) / k!
In this case, lambda=0.02 and t=50, and we want to find P(2).
Using the formula, we can substitute the values:
P(2) = (0.02^2 * e^(-0.02)) / 2!
Calculating this:
P(2) = (0.0004 * e^(-0.02)) / 2
Using a calculator or software to evaluate e^-0.02, the calculation can be done as follows:
P(2) = (0.0004 * 0.9801986733) / 2
P(2) = 0.0001960794 / 2
P(2) = 0.0000980397
Therefore, P(2) is approximately 0.0000980397.
To compute P(2) using the given values of lambda (λ) and t (time), we need to use the Poisson distribution formula. The Poisson distribution is used to model the probability of a certain number of events occurring in a fixed interval of time or space.
The formula for the Poisson distribution is:
P(x) = (e^(-λ) * λ^x) / x!
Where:
P(x) is the probability of x events occurring
e is the base of the natural logarithm (approximately 2.71828)
λ (lambda) is the mean number of events that occur in the given interval
x is the number of events we want to calculate the probability for
x! represents the factorial of x
In this case, we have λ = 0.02 and we want to compute P(2) for x = 2.
So, substituting the values into the formula, we have:
P(2) = (e^(-0.02) * 0.02^2) / 2!
Now, let's calculate it step by step:
1. Calculate e^(-0.02):
Using a calculator or math library, calculate e^(-0.02) = 0.98019867 (approx)
2. Calculate 0.02^2:
0.02^2 = 0.0004
3. Calculate 2! (factorial of 2):
2! = 2 * 1 = 2
4. Substitute the values into the formula:
P(2) = (0.98019867 * 0.0004) / 2
5. Simplify the expression:
P(2) = 0.0003920793367
Therefore, P(2) is approximately 0.0003920793367