Discussion6

Martin Luo

2023 10 16

1. Normal Distribution (Gaussian Distribution):

normal distribution is a fundamental probability distribution featuring a symmetric bell-shaped curve centered around the mean. It is defined by two crucial parameters: the mean, which establishes the distribution’s midpoint, and the standard deviation, which quantifies the data’s variability. Within a normal distribution, about 68% of data points lie within one standard deviation from the mean, 95% within two standard deviations, and approximately 99.7% within three standard deviations.

2. Uniform Distribution:

A uniform distribution is a fundamental probability distribution in which each value within a specific range has an equal likelihood of occurring. The probability density function of this distribution is represented by a horizontal line, signifying equal probabilities for all values within that range.

3.Binomial Distribution:

The binomial distribution is a discrete probability distribution applicable to independent repeated experiments with two possible outcomes. It is determined by two key parameters: the probability of success (p) and the number of trials (n). This distribution describes the number of successful events occurring in n independent repeated trials.

PDF, or Probability Density Function, is used to describe the probability distribution of the occurrence of values for a continuous random variable within a specific range. The horizontal axis of the PDF represents the values of the random variable, while the vertical axis indicates the likelihood of those values occurring. For continuous random variables, the integral of the PDF represents the probability of the random variable taking a value less than or equal to a certain real number.

CDF, or Cumulative Distribution Function, describes the probability that a discrete random variable takes a value less than or equal to a given real number. In other words, the CDF is the integral form of the PDF, with its vertical axis representing the probability of values for a discrete random variable. For continuous random variables, the CDF is the integral of the PDF.

The key parameters that define the Normal Distribution are the mean (μ) and the standard deviation (σ). These parameters determine the center and the spread of the distribution, respectively.

The key parameters that define the Binomial Distribution.The number of trials (n): This represents the total number of independent and identical trials or experiments.The probability of success in each trial (p): This is the probability of a successful outcome in each individual trial.

The key parameters that define the Uniform Distribution Lower bound (a): This is the minimum value within the range of possible values.Upper bound (b): This is the maximum value within the range of possible values.

Normal Distribution: For example exam scores In situations where student performance is influenced by a multitude of factors.

Binomial Distribution: For example coin flips; The outcome of a series of coin flips, where success might be defined as getting heads and failure as getting tails, can be modeled using a Binomial distribution.

Uniform Distribution: For exmaple dice rolls; The result of rolling a six-sided die is typically a Uniform Distribution, with each face of the die having an equal probability of landing face up.

install.packages("ggplot2", repos = "https://cran.r-project.org")

## 将程序包安装入'C:/Users/pokem/AppData/Local/R/win-library/4.3'
## (因为'lib'没有被指定)

## 程序包'ggplot2'打开成功，MD5和检查也通过
## 
## 下载的二进制程序包在
##  C:\Users\pokem\AppData\Local\Temp\RtmpaGYfoV\downloaded_packages里

library(ggplot2)
x <- seq(-3, 3, by = 0.01)
pdf_values <- dnorm(x)
cdf_values <- pnorm(x)
plot(x, pdf_values, type = "l", col = "red", xlab = "X", ylab = "PDF",
     main = "PDF of Standard Normal Distribution")

plot(x, cdf_values, type = "l", col = "blue", xlab = "X", ylab = "CDF",
     main = "CDF of Standard Normal Distribution")

Part 2

we let number of procedures(N)= 100

total death resulted(x)= 10

success rate pi= 0.5

Binomial

N <- 100
x <- 10
p <- 0.05
prob <- dbinom(x, size = N, prob = p)
prob

## [1] 0.01671588

Poisson Model

lambda <- N * p
dpois(x, lambda)

## [1] 0.01813279