Probability theory ~ Poisson distribution

May be the reason for the holiday or because of today's walk fell, I think my arm is particularly painful, typing is particularly uncomfortable, feel particularly sleepy, see a good long time probability of the problem I do not understand.

Let's write and analyze. when n is very large p is very small, the two-item distribution C (N,K) =PK (1-p) n-k approximate to the usual when n≧10,p≦0.1, can be calculated by the Poisson formula approximation.

The parameter λ of the Poisson distribution is the average occurrence of random events in the unit time (or unit area). The Poisson distribution is suitable for describing the number of random events that occur per unit time.

Λ=NP, the average number of rare events that occurred in n experiments

The expectation and variance of Poisson distribution are λ

Application Scenario editing In the actual case, when a random event, such ascalls received by a telephone exchange station、a passenger arriving at a bus stop、particles emitted by a radioactive substance、white blood cells in an area under the microscopeWait a minuteat a fixed average instantaneous rate λ(or density) appears randomly and independently，then the number or number of occurrences of this event within a unit time (area or volume) follows the Poisson distribution P( λ) nearest to。 Therefore, Poisson distribution plays an important role in management science, operational research and some problems of natural science. Here is an example

Suppose that the number of earthquakes occurring in any time interval of T (years) N (t) obeys the Poisson distribution with the parameter λ=0.1t , andX represents a continuous two earthquakes Interval time (in years).

(1) prove that x follows exponential distribution and find out the distribution function of x;

(2) The probability of another earthquake occurring in the next 3 years;

(3) The probability of another earthquake occurring in the next 3 to 5 years.

Answer:

(1) When t≥0, P{x>t}=p{n (t) =0}=e-0.1t,//that should have been in the T-time earthquake, but did not occur, then in this period of time, the number of earthquakes in the frequency of the 0,0 factorial of the 0 is 1????? Vaguely remember what it was like to make.

∴f (t) =p{x≤t}=1-p{x>t}=1-e-0.1t;

When t<0, F (t) = 0,

∴f (x) ={1-e-0.1t,t≥0

0,t<0,

The problem is to study the random distribution of random variable x, the distribution of this random variable is related to the value of T, when T is greater than 0 when the earthquake occurred in the T-time, when T-yu 0 indicates that no earthquake occurred

The distribution function of the occurrence of the earthquake is the distribution function of the random variable x<=t time, when x is greater than T, the earthquake is not occurred, i.e. n (t) = 0;

X follows exponential distribution (λ=0.1);

(2) F (3) =1-e-0.1x3≈0.26;

(3) F (5)-F (3) ≈0.13.

Why do I think this problem is super difficult to understand, there is a particularly big knocking on the keyboard sound, really good annoying ah

Probability theory ~ Poisson distribution