Poison Distribtion problems
Example: Customers call us at a rate of 12 per minute. What is the probability of having exactly twenty customers call us within the span of a minute?
You should use R’s dpois function. You can use this to calculate the probability of getting X events within a period where the rate is Zs. Example code
below:
# dpois r - calculate poisson distribution probability in r
dpois(20, lambda=12)
[1] 0.009682032
Example: Customers call us at a rate of 12 per minute. “The boss” wants us to deliver excellent service and stay very productive. Our service will suffer
if we get more than twenty calls in a minute. We’re going to look lazy if five or less calls arrive in a minute. What are the odds of getting in trouble
with the boss?
For this problem, we’re going to use R’s ppois function, which gives the cumulative probability of an event. This is a digital version of the table of
probabilities included as an appendix in your favorite statistics book. It includes the option of specifying if we’re interested in the upper or lower tail of
the statistical distribution.
# simulating poisson process r
# cumulative poisson distribution
# ppois r - odds of more than 20 people calling
# default setting uses lower tail of distribution
ppois(20, lambda = 12,lower.tail=FALSE)
[1 ] 0.01159774
# ppois r - odds of 5 or less people calling
ppois(5, lambda = 12)
[1] 0.02034103
3If there are twelve cars crossing a bridge per minute on average, find
the probability of having seventeen or more cars crossing the bridge in
a particular minute.
Solution
The probability of having sixteen or less cars crossing the bridge in a
particular minute is given by the function ppois.
> ppois(16, lambda=12) # lower tail
[1] 0.89871
Hence the probability of having seventeen or more cars crossing the
bridge in a minute is in the upper tail of the probability density
function.
> ppois(17 , lambda=12, lower.tail=FALSE) # upper tail
[1] 0.10129
The Poisson Distribution in R
Michael Foley
January 8, 2019
R Functions dpois, ppois, and rpois
Random varaible XX is distributed X∼P(λ)X∼P(λ) with
mean μ=λμ=λ and variance σ2=λσ2=λ if X=xX=x is the
number of successes in nn (many) trials when the
probability of success λ/nλ/n is small. The probability
of X=kX=k successes is Pr(X=k)=(e−λλk)/k!
Pr(X=k)=(e−λλk)/k!.
R function dpois(x, lambda) is the probability
of x successes in a period when the expected number of
events is lambda. R function ppois(q, lambda,
lower.tail) is the cumulative probability (lower.tail
= TRUE for left tail, lower.tail = FALSE for right tail)
of less than or equal to q successes. R function rpois(n,
lambda) returns n random numbers from the Poisson
distribution x ~ P(lambda). R function qpois(p,
lambda, lower.tail returns the value (quantile) at the
specified cumulative probability (percentile) p.
Example
What is the probability of making 2 to 4 sales in a week if
the average sales rate is 3 per week?
# Using cumulative probability
ppois(q = 4, lambda = 3, lower.tail = TRUE) -
ppois(q = 1, lambda = 3, lower.tail = TRUE)
## [1] 0.616115
# Using exact probability
dpois(x = 2, lambda = 3) +
dpois(x = 3, lambda = 3) +
dpois(x = 4, lambda = 4)
## [1] 0.6434504
# expected number of sales = lambda = 3
# variance = lambda = 3
library(ggplot2)
library(dplyr)
options(scipen = 999, digits = 2) # sig digits
events <- 0:10
density <- dpois(x = events, lambda = 3)
prob <- ppois(q = events, lambda = 3, lower.tail = TRUE)
df <- data.frame(events, density, prob)
ggplot(df, aes(x = factor(events), y = density)) +
geom_col() +
geom_text(
aes(label = round(density,2), y = density + 0.01),
position = position_dodge(0.9),
size = 3,
vjust = 0
) +
labs(title = "PMF and CDF of Poisson Distribution",
subtitle = "P(3).",
x = "Events (x)",
y = "Density") +
geom_line(data = df, aes(x = events, y = prob))
Example
Suppose a baseball player has a p=.300 batting average. What is the probability of X<=150 hits in n=500 at bats? X=150? X>150?
# probability of x <= 150
ppois(q = 150, lambda = .300 * 500, lower.tail = TRUE)
## [1] 0.52
# probability of x = 150
dpois(x = 150, lambda = .300 * 500)
## [1] 0.033
# probability of x > 150
ppois(q = 150, lambda = .300 * 500, lower.tail = FALSE)
## [1] 0.48
library(ggplot2)
library(dplyr)
options(scipen = 999, digits = 2) # sig digits
hits <- 0:100 * 3
density <- dpois(x = hits, lambda = .300 * 500)
prob <- ppois(q = hits, lambda = .300 * 500, lower.tail = TRUE)
df <- data.frame(hits, density, prob)
ggplot(df, aes(x = hits, y = density)) +
geom_col() +
labs(title = "Poisson(150)",
subtitle = "PMF and CDF of Poisson(3) distribution.",
x = "Hits (x)",
y = "Density") +
geom_line(data = df, aes(x = hits, y = prob))
The Poisson distribution approximates the binomial distribution
with λ=npλ=np if n>=20n>=20 and p<=0.05p<=0.05.
Example
What is the distribution of successes from a sample of n = 50 when the probability of success is
p = .03?
library(ggplot2)
library(dplyr)
library(tidyr)
## Warning: package 'tidyr' was built under R version 3.4.4
options(scipen = 999, digits = 2) # sig digits
n = 0:10
df <- data.frame(events = 0:10,
Poisson = dpois(x = n, lambda = .03 * 50),
Binomial = dbinom(x = n, size = 50, p = .03))
df_tidy <- gather(df, key = "Distribution", value = "density", -c(events))
ggplot(df_tidy, aes(x = factor(events), y = density, fill = Distribution)) +
geom_col(position = "dodge") +
labs(title = "Poisson(15) and Binomial(50, .03)",
subtitle = "Poisson approximates binomial when n >= 20 and p <= .05.",
x = "Events (x)",
y = "Density")
Example
Suppose the probability that a drug produces a certain side effect is p =
= 0.1% and n = 1,000 patients in a clinical trial receive the drug. What
is the probability 0 people experience the side effect?
# The expected value is np
1000 * .001
## [1] 1
# The probability of measuring 0 when the expected value is 1
dpois(x = 0, lambda = 1000 * .001)
## [1] 0.37
library(ggplot2)
library(dplyr)
options(scipen = 999, digits = 2) # sig digits
x <- 0:10
density <- dpois(x = x, lambda = 1000 * .001)
prob <- ppois(q = x, lambda = 1000 * .001, lower.tail = TRUE)
df <- data.frame(x, density, prob)
ggplot(df, aes(x = x, y = density)) +
geom_col() +
geom_text(
aes(label = round(density,2), y = density + 0.01),
position = position_dodge(0.9),
size = 3,
vjust = 0
) +
labs(title = "Poisson(1)",
subtitle = "PMF and CDF of Poisson(1) distribution.",
x = "Events (x)",
y = "Density") +
geom_line(data = df, aes(x = x, y = prob))
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