#A: Normal Distribution: It is continuous probability distribution which parameter mean and deviation. The data usually fall as bell shape and it is broadly used in many areas such as economics, finance etc.

Binomial Distribution: It is discrete probability distribution and it a modification from Bernoulli Distribution. It represents the number of successses in a discrete number of independent trails. Two parameters: n and p where n is number of trails and p is probability of success.

Poisson Distribution: It is also discrete probability distribution and it represents number of the events happens in certain period. Parameter is lamda which is the average number of occurences. It can convert to distributions such as binomial or normal in some cases.

#B: PDF measures the probability of a certain value and CDF measures the cumulative probability up to a certain value. Binomial as example: PMF:P(X=k)=(n choose k)p^k (1-p)^(n-k) n choose k chooses favor outcomes. p^k is probability of favor outcomes and (1-p)^(n-k) is probabily of infavor outcomes which makes sense. CMF: ∑ (i=0,k) (n choose k)p^k (1-p)^(n-k) which sums up the PMF from previous also makes sense.

#C: Normal: mean (μ), standard deviation (σ) Binominal: Number trails(n), Probability of success(p) Poisson: Average number of occurrences within certain time(λ) Yes, R requires these key parameters.

#D: Normal: The stock price may follow normal distribution. It’s also used as regression assumptions. Binominal: Tossing coins, number of head appears. Poisson: Number of airplanes arrived in one day follows poisson distribution.

#E:
# Loading necessary libraries
library(ggplot2)

# Parameters for the Poisson distribution
lambda <- 5  # mean number of occurrences

# Generating the Poisson distribution
x <- 0:15  # considering 16 points as a reasonable range for visualization
y <- dpois(x, lambda)

# Plotting the distribution using ggplot2
df <- data.frame(x, y)
p <- ggplot(df, aes(x=x, y=y)) + 
  geom_bar(stat="identity", fill="red") + 
  labs(title="Poisson Distribution", x="Number of occurrences", y="Probability") +
  theme_minimal()
print(p)

# Part 2
# Parameters
N <- 100  # Number of procedures
x <- 15    # Number of deaths
pi <- 0.05  # Probability of death within 30 days
#Bio
dbinom(x, size = N, prob = pi)
## [1] 9.880016e-05
#mean for Poisson distribution
Q2_mean = N*pi
#Poisson distribution
dpois(x, Q2_mean)
## [1] 0.0001572454