PART I

A.

Normal Distribution: it is a continuous probability distribution characterized by a symmetric bell-shaped curve, defined by two parameters: the mean \((μ)\), which determines the location of the peak, and the standard deviation \((σ)\), which measures the spread or width of the curve.

Binomial Distribution: it is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success \((p)\). The distribution is characterized by two parameters: the number of trials \((n)\) and the probability of success \((p)\), and it provides the probability of observing a particular number of successes \((k)\) in n trials.

Poisson Distribution: it is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given the average number of times the event occurs over that interval. It is characterized by the parameter \(λ\), which is the mean number of occurrences in the interval.

B.

PDF: The Probability Density Function gives you the probability of a random variable falling within a particular range of values. For discrete distributions, like the binomial distribution, it tells us the probability of obtaining a specific outcome.

CDF: The Cumulative Distribution Function gives the probability that a random variable takes on a value less than or equal to a particular point. It is essentially the cumulative sum of the probabilities from the PDF, starting from the lowest possible value up to the value of interest.

Pick Binomial Distribution.

The PDF for the binomial distribution is given by: \[P(X=k)={n\choose k}p^k(1-p)^{n-k}\] Intuition Behind the PDF:

In \(n\) trials, there are \({n\choose k}\) ways to choose \(k\) successful outcomes. \({n\choose k}\) accounts for all possible orders in which successes can occur.

\(p^k\) represents the likelihood of achieving exactly \(k\) successes, ignoring failures for a moment.

\((1-p)^{n-k}\) takes into account the likelihood of the remaining \(n-k\) outcomes being failures.

Together, the formula essentially calculates all possible ways of obtaining \(k\) successes and multiplies this by the probability of obtaining those successes and the remaining failures in \(n\) trials.

The CDF for a binomial distribution is given by: \[F(X)=P(X{\leq}x)=\sum^{X}_{k=0}P(X=k)\] Intuition Behind the CDF:

The CDF gives us a running total of probabilities up to a certain point \(x\). It helps us answer questions like “What’s the probability of observing \(x\) or fewer successes in \(n\) trials?”. In practical terms, the CDF helps in understanding the likelihood that a random variable \(X\) will take a value less than or equal to a certain point \(x\).

C.

Normal Distribution: Mean (\(μ\)) and Standard Deviation (\(σ\))

Binomial Distribution: Number of Trials (\(n\)) and Probability of Success (\(p\))

Poisson Distribution: Rate (\(λ\))

D.

Normal Distribution:

Human Characteristics are always related to Normal Distribution like height and IQ Scores.

Height: If you measure the heights of a large group of individuals, the distribution of heights tends to be normally distributed.

IQ Scores: IQ scores are typically distributed normally around an average score of 100.

Binomial Distribution:

Quality Control

If you’re producing items and each item can either be defective or not defective, the number of defective items in a sample can be modeled with a binomial distribution.

Poisson Distribution:

Customers Arrival

The number of customers arriving at a store in a given time interval can be modeled using a Poisson distribution, assuming the arrival of one customer is independent of the arrival of another.

E.

Normal Distribution:

# Parameters
mu <- 0
sigma <- 1

# Data
x <- seq(-5, 5, by = 0.1)
y <- dnorm(x, mean=mu, sd=sigma)

# Plot
plot(x, y, type="l", main="Normal Distribution", xlab="x", ylab="Density")

Binomial Distribution:

# Parameters
n <- 10
p <- 0.5

# Data
x <- 0:n
y <- dbinom(x, size=n, prob=p)

# Plot
plot(x, y, type="h", main="Binomial Distribution", xlab="x", ylab="Probability", lwd=2)
points(x, y, pch=16, col="red")

Poisson Distribution:

# Parameters
lambda <- 5

# Data
x <- 0:15
y <- dpois(x, lambda=lambda)

# Plot
plot(x, y, type="h", main="Poisson Distribution", xlab="x", ylab="Probability", lwd=2)
points(x, y, pch=16, col="red")

PART II

\[N=100\] \[x=15\] \[π=0.1\] Binomial Distribution

plot(x = 0:25, y = dbinom(x = 0:25, size = 100, prob = 0.1), type = 'h', main = 'Binomial Distribution', xlab = 'Deaths', ylab = 'Probability', lwd = 2)

Poisson Distribution

plot(x = 0:25, y = dpois(x = 0:25, lambda = 10), type = 'h',main = 'Poisson Distribution', xlab = 'Deaths', ylab = 'Probability', lwd = 2)

These distributions look about identical.

The Poisson distribution can be thought of as a limiting case of the binomial distribution as \(n\) (the number of trials) approaches infinity and \(p\) (the probability of success) approaches zero, such that \(np\) (the expected number of successes) remains constant. Specifically, if \(n\) is large and p is small such that \(np=λ\) is moderate, then the binomial distribution can be approximated by a Poisson distribution with parameter \(λ\).