Poisson distribution table

Poisson distribution cumulative probability table P(X ≤ k) for various values of λ. Interactive calculator included.

k  λ 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0
0 0.60653 0.36788 0.22313 0.13534 0.08208 0.04979 0.03020 0.01832 0.01111 0.00674 0.00248 0.00091 0.00034 0.00012 0.00005
1 0.90980 0.73576 0.55783 0.40601 0.28730 0.19915 0.13589 0.09158 0.06110 0.04043 0.01735 0.00730 0.00302 0.00123 0.00050
2 0.98561 0.91970 0.80885 0.67668 0.54381 0.42319 0.32085 0.23810 0.17358 0.12465 0.06197 0.02964 0.01375 0.00623 0.00277
3 0.99825 0.98101 0.93436 0.85712 0.75758 0.64723 0.53663 0.43347 0.34230 0.26503 0.15120 0.08177 0.04238 0.02123 0.01034
4 0.99983 0.99634 0.98142 0.94735 0.89118 0.81526 0.72544 0.62884 0.53210 0.44049 0.28506 0.17299 0.09963 0.05496 0.02925
5 0.99999 0.99941 0.99554 0.98344 0.95798 0.91608 0.85761 0.78513 0.70293 0.61596 0.44568 0.30071 0.19124 0.11569 0.06709
6 1.00000 0.99992 0.99907 0.99547 0.98581 0.96649 0.93471 0.88933 0.83105 0.76218 0.60630 0.44971 0.31337 0.20678 0.13014
7 1.00000 0.99999 0.99983 0.99890 0.99575 0.98810 0.97326 0.94887 0.91341 0.86663 0.74398 0.59871 0.45296 0.32390 0.22022
8 1.00000 1.00000 0.99997 0.99976 0.99886 0.99620 0.99013 0.97864 0.95974 0.93191 0.84724 0.72909 0.59255 0.45565 0.33282
9 1.00000 1.00000 1.00000 0.99995 0.99972 0.99890 0.99669 0.99187 0.98291 0.96817 0.91608 0.83050 0.71662 0.58741 0.45793
10 1.00000 1.00000 1.00000 0.99999 0.99994 0.99971 0.99898 0.99716 0.99333 0.98630 0.95738 0.90148 0.81589 0.70599 0.58304
11 1.00000 1.00000 1.00000 1.00000 0.99999 0.99993 0.99971 0.99908 0.99760 0.99455 0.97991 0.94665 0.88808 0.80301 0.69678
12 1.00000 1.00000 1.00000 1.00000 1.00000 0.99998 0.99992 0.99973 0.99919 0.99798 0.99117 0.97300 0.93620 0.87577 0.79156
13 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99998 0.99992 0.99975 0.99930 0.99637 0.98719 0.96582 0.92615 0.86446
14 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99998 0.99993 0.99977 0.99860 0.99428 0.98274 0.95853 0.91654
15 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99998 0.99993 0.99949 0.99759 0.99177 0.97796 0.95126
16 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99999 0.99998 0.99983 0.99904 0.99628 0.98889 0.97296
17 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99999 0.99994 0.99964 0.99841 0.99468 0.98572
18 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99998 0.99987 0.99935 0.99757 0.99281
19 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99999 0.99996 0.99975 0.99894 0.99655
20 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99999 0.99991 0.99956 0.99841

What is the Poisson distribution?

The Poisson distribution models the number of events occurring in a fixed interval of time or space when events occur independently at a constant average rate \(\lambda\). Examples: number of calls per hour, number of defects per unit.

How to use this table

The table gives \(P(X \leq k \mid \lambda)\) — the probability that a Poisson random variable with mean \(\lambda\) is at most \(k\).

  • \(P(X = k) = P(X \leq k) - P(X \leq k-1)\)
  • \(P(X > k) = 1 - P(X \leq k)\)
  • \(P(a \leq X \leq b) = P(X \leq b) - P(X \leq a-1)\)

Worked example

A call centre receives an average of \(\lambda = 3\) calls per minute. What is the probability of receiving at most 2 calls in one minute?

Row k = 2, column λ = 3.0 → \(P(X \leq 2 \mid \lambda = 3) = 0.42319\).