P(X>15)=1−∑ k=015​ k!e −10 ⋅10 k

It seems like you're asking for the probability that a random variable X is greater than 15, given that X follows a Poisson distribution with a mean (λ) of 10. The formula you've provided is the cumulative distribution function (CDF) for a Poisson distribution, but it's not quite correct. Here's the correct formula and how to use it:

The probability mass function (PMF) for a Poisson distribution is:

P(X=k) = λ^k * e^-λ / k!

where:

• λ is the mean of the distribution,
• k is the specific value you're interested in,
• e is the base of the natural logarithm (approximately 2.71828),
• and "!" denotes the factorial function.

The cumulative distribution function (CDF), which gives the probability that X is less than or equal to a certain value, is the sum of the PMFs for all values up to that value. So, for a Poisson distribution:

P(X≤k) = ∑ (from i=0 to k) [λ^i * e^-λ / i!]

To find P(X>15), you want the complement of P(X≤15), which is 1 - P(X≤15). So you would calculate:

P(X>15) = 1 - ∑ (from i=0 to 15) [10^i * e^-10 / i!]

This sum can be calculated manually for each value of i from 0 to 15, but it's often easier to use statistical software or a scientific calculator that can handle factorials and exponentials.