statistics 1B

Probability distributions are formal mathematical descriptions of the likelihood of different outcomes for a random variable. They are classified into discrete and continuous types based on the nature of their random variable's support.

A discrete random variable $X$ has a countable number of possible outcomes. Its probability distribution is defined by a Probability Mass Function (PMF), $P(X=x)$, where $x$ is a specific outcome.

• Expected Value of the Sample Mean:
  $E[\bar{X}] = E\left[\frac{1}{n} \sum_{i=1}^{n} X_i\right] = \frac{1}{n} \sum_{i=1}^{n} E[X_i] = \frac{1}{n} \sum_{i=1}^{n} \mu = \frac{n\mu}{n} = \mu$
  This demonstrates that $\bar{X}$ is an unbiased estimator of $\mu$.

Hypothesis testing, at its core, is a formal procedure for making an informed decision about a population parameter based on sample data. The framework begins with the formulation of two competing hypotheses: the null hypothesis ($H_0$) and the alternative hypothesis ($H_1$).

Hypothesis testing for two samples primarily involves comparing parameters (means, proportions, variances) from two distinct populations based on sample data. The fundamental principle remains the construction of a null hypothesis ($H_0$), representing no difference, and an alternative hypothesis ($H_1$ or $H_a$), representing a significant difference.

Simple Linear Regression (SLR) models the relationship between two continuous quantitative variables: a dependent variable, $Y$, and an independent variable, $X$. This relationship is assumed to be linear in its parameters. Correlation quantifies the strength and direction of the linear association between these variables.

To find $\hat{\beta}_0$ and $\hat{\beta}_1$, we take partial derivatives of $SSR$ with respect to $\hat{\beta}_0$ and $\hat{\beta}_1$, set them to zero, and solve the resulting system of "normal equations."