715. Poisson Random Variable - JulTob/Mathematics GitHub Wiki
A Poisson random variable describes a random experiment where the variable represents the number of times a certain event occurs (i.e., it takes integer values) within a given time interval. The length of this interval is irrelevant; it is considered a generic measure.
- Rate of occurrence: α=5\alpha = 5
Each value has a specific probability of occurrence:
| Variable value kk | Probability pX(k)p_X(k) | kpX(k)k p_X(k) | k2pX(k)k^2 p_X(k) |
|---|---|---|---|
| 0 | 0.006737947 | 0 | 0 |
| 1 | 0.033689735 | 0.033689735 | 0.033689735 |
| 2 | 0.08422433749 | 0.168448675 | 0.33689735 |
| 3 | 0.1403738958 | 0.4211216874 | 1.263365062 |
| 4 | 0.1754673698 | 0.7018694791 | 2.807477916 |
| 5 | 0.1754673698 | 0.8773368488 | 4.386684244 |
| 6 | 0.1462228081 | 0.8773368488 | 5.264021093 |
| 7 | 0.104444863 | 0.7311140407 | 5.117798285 |
| 8 | 0.06527803935 | 0.5222243148 | 4.177794518 |
| 9 | 0.03626557742 | 0.3263901967 | 2.937511771 |
| 10 | 0.01813278871 | 0.1813278871 | 1.813278871 |
| 11 | 0.008242176685 | 0.09066394354 | 0.9973033789 |
| 12 | 0.003434240286 | 0.04121088343 | 0.4945306011 |
| 13 | 0.001320861648 | 0.01717120143 | 0.2232256186 |
| 14 | 0.000471736303 | 0.006604308241 | 0.09246031538 |
| 15 | 0.0001572454343 | 0.002358681515 | 0.03538022272 |
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title "Poisson Distribution (α=5)"
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- The variable represents the count of events occurring in a fixed interval.
- The probability of multiple occurrences within small subintervals is determined by α\alpha.
- The expectation and variance are both equal to α\alpha.
The Poisson random variable approximates the binomial distribution when the probability of success in a binomial experiment is very small, and the number of trials is very large. In this case, the Poisson variable's parameter α\alpha is equal to the product of nn and pp from the binomial distribution.
This type of random variable is widely used in:
- Queueing theory, modeling arrival rates of customers, calls, or events.
- Reliability engineering, predicting system failures over time.
- Biology and epidemiology, modeling occurrences of rare diseases or mutations.
- Finance and insurance, predicting claim frequencies and risk assessments.
