Topic 3 – Discrete Distributions
ENVX1002 Introduction to Statistical Methods
Jan 2024
Example
- We have 5 insects which we spray with an insecticide, each insect has a 60% chance of being killed;
- P(K) = 0.6
- Possible questions:
- What is the probability that all 5 insects will be killed?
- What is the probability that at least 3 insects will be killed?
- The data is ‘binary’ and the events are mutually exclusive (either dead or alive unless it is zombie fly);
- we can say the data is categorical or numeric discrete
- we can use a binomial (discrete) distribution to “model” the data.
Example plot
Example
- How many combinations are there for exactly 2 flies to die out of 5 flies?
{5\choose{2}}=\frac{5!}{2!(5−2)!}= \frac{(5×4×3×2×1)}{(2×1×3×2×1)}=10
- What is the probability that exactly 2 flies will die?
P(X=x)={n\choose{x}}p^x(1-p)^{n-x}={5\choose{2}}0.6^2(1-0.6)^{5-2}
=10×0.36×0.064=0.2304
Point, cumulative or interval probabilities
- We have already calculated a point probability i.e. the probability of exactly 2 flies dying.
- But what about if we wanted to know the probability of 2 or more flies dying P(X\ge{2}) or between 2 and 4 flies P(2\le{X}\le{4}) dying??
- So how can we calculate these?
- One way is to calculate all the point probabilities from 0-5 and add the probabilities cumulatively or in the interval
Point, cumulative or interval probabilities
P(X\ge{2})=(X=2)+(X=3)+(X=4)+(X=5)=0.2304+0.3456+0.2592+0.07776=0.91296
- You guys try to calculate the following
P(2\le{X}\le{4})=??
Cumulative
P(X\ge{2})
- =1-BINOM.DIST(1,5,0.6,TRUE)
Interval
P(2 \le X \le 4)
- =BINOM.DIST(4,5,0.6,TRUE)-BINOM.DIST(1,5,0.6,TRUE)
Count Data
- Often we are interested in count data such as the number of events occurring in an interval;
- We generally model this data using the Poisson distribution (Described by Simeon-Denis Poisson) where we use \lambda denotes the average number of events occurring in an interval
- We often write this as
X\sim{Po(\lambda)}
- Where X is the number of discrete and independent events in the interval
- e.g. number of plants of a certain species along transect
- e.g. occurrence of disease in a period of time
Example with kick deaths
Example with kick deaths
- So now you all can calculate what the probability is, as an example, the of having less than 2 deaths across all cavalry corps for the period of 1875-1894 P(X<2).
- =POISSON(1,0.7,TRUE)
Example with kick deaths
- Another example, is having exactly 2 deaths across all cavalry corps for the period of 1875-1894 P(X=2).
- =POISSON(2,0.7,FALSE)
Interesting results with the binomial and Poisson distributions
# Parameters for the binomial distribution
n <- 100
p <- 0.05
# Calculating lambda for the Poisson approximation
lambda <- n * p
# Generate the range of values
x <- 0:n
# Data frames for plotting
data_binom <- data.frame(x = x, probability = dbinom(x, n, p), Distribution = "Binomial")
data_pois <- data.frame(x = x, probability = dpois(x, lambda), Distribution = "Poisson")
# Combine data
data_combined <- rbind(data_binom, data_pois)
# Create the plot
p <- ggplot(data_combined, aes(x = factor(x), y = probability, fill = Distribution)) +
geom_bar(stat = "identity", position = position_dodge()) +
ggtitle("Binomial vs Poisson Distribution") +
xlab("Number of successes") +
ylab("Probability") +
scale_fill_manual(values = c("blue", "red")) +
theme_minimal() +
scale_x_discrete(breaks = seq(0, n, by = 10)) # Display every 10th label
Thanks!
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