Poisson normal distribution an Introduction:
Normal distributon:
In probability theory and statistics, the normal distribution or Gaussian distribution is an absolutely continuous probability distribution with zero cumulants of all orders higher than two.
Poisson distributon:
In probability theory and statistics, the Poisson distribution (or Poisson law of large numbers) is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event.
Poisson Distributon in Poisson and Normal Distribution:
The three basic assumptions on poisson distribution are as follows
The probability of one photon arriving in ?t is proportional to ?t when ?t is very small.
P(1;?t) = a?t for small ?t
where a is a constant whose value is not yet determined.
The probability that more than one photon arrives in ?t is negligible when ?t is very small.
P(0;?t)+P(1;?t) = 1 for small ?t
The number of photons that arrive in one interval is independent of the number of photons that arrive in any other non-overlapping interval.
Variance of the Poisson Distribution:
var(k) = E[k2] - E2[k] = at
Note that for a poisson distribution mean and the variance are equal .
Algorithm for poisson distribution:
Algorithm :Poisson random number:
init:
Let L ? e-?, k ? 0 and p ? 1.
do:
Generate uniform random number u in [0,1] and let p ? p × u.
while p > L.
return k - 1.
Normal Distribution in Poisson and Normal Distribution:
Description of normal distribution:
For any of situation in which the absolute value of a continuous variable is changed randomly from trial to trial.
The random improbability or random error.
Some properties of normal distribution Bell curve:
The curve will be Symmetric,
It will be unimodal,
The value will be extends to +/- infinity,
And the area under the curve is always one (it will be the fixed value).
The normal distribution bell curve is fully based on the mean and standard deviation functions.
These are some of the important points of poisson and normal distribution.
Normal distributon:
In probability theory and statistics, the normal distribution or Gaussian distribution is an absolutely continuous probability distribution with zero cumulants of all orders higher than two.
Poisson distributon:
In probability theory and statistics, the Poisson distribution (or Poisson law of large numbers) is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event.
Poisson Distributon in Poisson and Normal Distribution:
The three basic assumptions on poisson distribution are as follows
The probability of one photon arriving in ?t is proportional to ?t when ?t is very small.
P(1;?t) = a?t for small ?t
where a is a constant whose value is not yet determined.
The probability that more than one photon arrives in ?t is negligible when ?t is very small.
P(0;?t)+P(1;?t) = 1 for small ?t
The number of photons that arrive in one interval is independent of the number of photons that arrive in any other non-overlapping interval.
Variance of the Poisson Distribution:
var(k) = E[k2] - E2[k] = at
Note that for a poisson distribution mean and the variance are equal .
Algorithm for poisson distribution:
Algorithm :Poisson random number:
init:
Let L ? e-?, k ? 0 and p ? 1.
do:
Generate uniform random number u in [0,1] and let p ? p × u.
while p > L.
return k - 1.
Normal Distribution in Poisson and Normal Distribution:
Description of normal distribution:
For any of situation in which the absolute value of a continuous variable is changed randomly from trial to trial.
The random improbability or random error.
Some properties of normal distribution Bell curve:
The curve will be Symmetric,
It will be unimodal,
The value will be extends to +/- infinity,
And the area under the curve is always one (it will be the fixed value).
The normal distribution bell curve is fully based on the mean and standard deviation functions.
These are some of the important points of poisson and normal distribution.
No comments:
Post a Comment