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💻Advanced R Programming Unit 6 – Probability & Statistical Inference

Probability and statistical inference form the backbone of data analysis in R programming. These concepts allow us to quantify uncertainty, make predictions, and draw conclusions from data. From basic probability calculations to advanced hypothesis testing, R provides powerful tools for statistical modeling and inference. Statistical techniques in R enable us to estimate population parameters, test hypotheses, and build predictive models. By leveraging probability distributions, sampling methods, and inferential statistics, we can extract meaningful insights from data and make informed decisions in various fields like finance, healthcare, and marketing.

Study Guides for Unit 6

6.1

Probability distributions in R

5 min read

6.2

Sampling and hypothesis testing

5 min read

6.3

Confidence intervals and p-values

7 min read

6.4

ANOVA and regression analysis

4 min read

6.5

Bayesian inference with MCMC

5 min read

Key Concepts and Terminology

Probability measures the likelihood of an event occurring ranges from 0 to 1
Random variables map outcomes of random events to numerical values (discrete or continuous)
Probability distributions describe the probabilities of different outcomes for a random variable
- Discrete distributions include Bernoulli, binomial, and Poisson
- Continuous distributions include normal, exponential, and uniform
Expected value represents the average outcome of a random variable over many trials
Variance and standard deviation measure the spread or dispersion of a probability distribution
Statistical inference draws conclusions about a population based on a sample of data
Hypothesis testing evaluates claims or assumptions about a population using sample data
- Null hypothesis ( $H_0$ ) represents the default or status quo assumption
- Alternative hypothesis ( $H_a$ or $H_1$ ) represents the claim being tested

Probability Fundamentals in R

Probability calculations in R use logical operators and functions like
```
sum()
```
and
```
length()
```
Generate random numbers from specific distributions using functions like
```
rnorm()
```
and
```
rbinom()
```
Calculate probabilities of events using the
```
prob()
```
function from the
```
prob
```
package
Compute expected values and variances of random variables using
```
E()
```
and
```
Var()
```
functions
Simulate random processes and estimate probabilities through repeated sampling
Visualize probability distributions using histograms (
```
hist()
```
), density plots (
```
plot(density())
```
), and cumulative distribution functions (
```
plot(ecdf())
```
)
Perform probability calculations on vectors and matrices using element-wise operations

Random Variables and Distributions

Discrete random variables have countable outcomes (integers) while continuous random variables have uncountable outcomes (real numbers)
Probability mass functions (PMFs) define probabilities for discrete random variables
Probability density functions (PDFs) define probabilities for continuous random variables
Cumulative distribution functions (CDFs) give the probability of a random variable being less than or equal to a specific value
Common discrete distributions in R include binomial (
```
dbinom()
```
), Poisson (
```
dpois()
```
), and geometric (
```
dgeom()
```
)
Common continuous distributions in R include normal (
```
dnorm()
```
), exponential (
```
dexp()
```
), and uniform (
```
dunif()
```
)
Use the
```
d
```
,
```
p
```
,
```
q
```
, and
```
r
```
prefixes for density, probability, quantile, and random generation functions respectively (e.g.,
```
dnorm()
```
,
```
pnorm()
```
,
```
qnorm()
```
,
```
rnorm()
```
)

Statistical Inference Techniques

Point estimation calculates a single value estimate of a population parameter from sample data (sample mean, sample proportion)
Interval estimation provides a range of plausible values for a population parameter (confidence intervals)
Maximum likelihood estimation (MLE) finds parameter values that maximize the likelihood of observing the sample data
Bayesian inference updates prior beliefs about parameters based on observed data to obtain posterior distributions
Resampling methods like bootstrapping and permutation tests estimate sampling distributions and p-values
Central Limit Theorem states that the sampling distribution of the mean approaches a normal distribution as sample size increases
Confidence intervals provide a range of values that likely contain the true population parameter with a specified level of confidence (95%)

Hypothesis Testing in R

Specify null and alternative hypotheses based on the research question or claim being investigated
Choose an appropriate test statistic and calculate its value from the sample data (t-statistic, z-statistic, chi-square statistic)
Determine the p-value associated with the test statistic under the null distribution
Make a decision to reject or fail to reject the null hypothesis based on the p-value and significance level ( $\alpha$ )
Conduct one-sample tests (
```
t.test()
```
), two-sample tests (
```
t.test()
```
,
```
var.test()
```
), and ANOVA (
```
aov()
```
) for comparing means
Perform proportion tests (
```
prop.test()
```
), chi-square tests (
```
chisq.test()
```
), and Fisher's exact test (
```
fisher.test()
```
) for categorical data
Interpret test results, effect sizes, and confidence intervals in the context of the research problem

Advanced Probability Models

Markov chains model systems transitioning between states with probabilities depending only on the current state
Poisson processes describe the occurrence of rare events over time or space with a constant average rate
Bayesian networks represent probabilistic relationships among variables using directed acyclic graphs (DAGs)
Stochastic processes are sequences of random variables evolving over time (random walks, Brownian motion)
Queuing theory analyzes waiting lines and service systems using probability distributions for arrival and service times
Reliability theory models the failure probabilities and lifetimes of components and systems
Simulation techniques like Monte Carlo methods estimate complex probabilities and distributions through repeated random sampling

Data Visualization for Probability

Histograms display the frequency or density of continuous or discrete variables (
```
hist()
```
)
Density plots estimate the probability density function of a continuous variable (
```
plot(density())
```
)
Box plots summarize the distribution of a variable using quartiles and outliers (
```
boxplot()
```
)
Scatter plots show the relationship between two continuous variables (
```
plot()
```
)
Bar charts compare frequencies or proportions of categorical variables (
```
barplot()
```
)
Mosaic plots visualize contingency tables and associations between categorical variables (
```
mosaicplot()
```
)
Heatmaps display patterns and relationships in two-dimensional data using color intensities (
```
heatmap()
```
)

Practical Applications in R

Conduct A/B testing to compare conversion rates or performance metrics between two groups
Analyze survey data to estimate population proportions and opinions with confidence intervals
Model customer churn or attrition using logistic regression and predict future churn probabilities
Assess the reliability and failure rates of products or systems using survival analysis techniques
Optimize inventory levels and order quantities using probability distributions for demand and lead times
Evaluate the performance of machine learning models using cross-validation and hypothesis tests
Simulate and analyze queuing systems to optimize resource allocation and minimize waiting times
Estimate the value at risk (VaR) of financial portfolios using Monte Carlo simulations and extreme value theory