Probability and Statistics Lab 4 Part 2

Info

Objectives

• Understand and apply Binomial and Poisson distributions using R.

Prerequisites

• Binomial and Poisson’s distribution using R Studio
• Probability and Statistics Lab 4 Part 1
• Probability and Statistics Lecture 13
• P4-part2.pdf

Solutions to Probability Problems Using R

Problem 1: Binomial Distribution

Given:

• Probability of graduating, $p=0.5$
• Number of students, $n=10$

We need to find the probability of:

1. None graduating.
2. One graduating.
3. At least one graduating.

# Parameters
n <- 10
p <- 0.5
 
# i. Probability that none will graduate
prob_none <- dbinom(0, size = n, prob = p)
 
# ii. Probability that one will graduate
prob_one <- dbinom(1, size = n, prob = p)
 
# iii. Probability that at least one will graduate
prob_at_least_one <- 1 - prob_none
 
# Print results
cat("Probability that none will graduate:", prob_none, "\n")
cat("Probability that one will graduate:", prob_one, "\n")
cat("Probability that at least one will graduate:", prob_at_least_one, "\n")

Problem 2: Binomial Distribution with Given Mean and Variance

Given:

• Mean ($\mu$) = 5
• Variance (${\sigma }^2$) = $\frac{10}{3}$

We need to find the binomial distribution parameters $n$ and $p$.

Using the formulas:

• $\mu =np$
• ${\sigma }^2=np(1-p)$

# Given values
mean_value <- 5
variance_value <- 10/3
 
# Solve for n and p
p <- 1 - (variance_value / mean_value)
n <- mean_value / p
 
# Print results
cat("n =", n, "\n")
cat("p =", p, "\n")
 
# Plot probability distribution and cumulative distribution
x <- 0:n
probabilities <- dbinom(x, size = n, prob = p)
cumulative_probabilities <- pbinom(x, size = n, prob = p)
 
# Plot probability distribution
barplot(probabilities, names.arg = x, main = "Probability Distribution", xlab = "X", ylab = "P(X)")
 
# Plot cumulative distribution
plot(x, cumulative_probabilities, type = "s", main = "Cumulative Distribution", xlab = "X", ylab = "F(X)", ylim = c(0, 1))
points(x, cumulative_probabilities, pch = 19)

Problem 3: Poisson Distribution

Given:

• Mean ($\lambda$) = 7.6

We need to find the probability of:

1. Less than three accidents.
2. Exactly three accidents.

# Parameters
lambda <- 7.6
 
# i. Probability of less than three accidents
prob_less_than_three <- ppois(2, lambda = lambda)
 
# ii. Probability of exactly three accidents
prob_exactly_three <- dpois(3, lambda = lambda)
 
# Print results
cat("Probability of less than three accidents:", prob_less_than_three, "\n")
cat("Probability of exactly three accidents:", prob_exactly_three, "\n")

Problem 4: Random Values from Binomial Distribution

Generate 8 random values from a sample of 150 with a probability of 0.4.

# Parameters
sample_size <- 150
probability <- 0.4
num_values <- 8
 
# Generate random values
random_values <- rbinom(num_values, size = sample_size, prob = probability)
 
# Print results
cat("Random values:", random_values, "\n")

Problem 5: Binomial Probability for Coin Tosses

Given:

• Number of tosses = 51
• Probability of heads = 0.5

We need to find:

1. The number of heads that will have a probability of 0.25.
2. The probability of getting 26 or fewer heads.

# Parameters
num_tosses <- 51
prob_heads <- 0.5
 
# i. Number of heads with probability of 0.25
# We need to find k such that P(X = k) is approximately 0.25
probabilities <- dbinom(0:num_tosses, size = num_tosses, prob = prob_heads)
heads_prob_025 <- which(probabilities >= 0.25)
 
# ii. Probability of getting 26 or fewer heads
prob_26_or_less <- pbinom(26, size = num_tosses, prob = prob_heads)
 
# Print results
cat("Number of heads with probability of 0.25:", heads_prob_025, "\n")
cat("Probability of getting 26 or fewer heads:", prob_26_or_less, "\n")