Mathematics Statistics and Probability Syllabus

State the general addition rule for two events.

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

When are two events mutually exclusive, and how does the addition rule simplify?

Two events are mutually exclusive (disjoint) when they cannot occur together, so $P(A \cap B) = 0$ and $P(A \cup B) = P(A) + P(B)$.

Define the conditional probability of $A$ given $B$.

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0$$

State the general multiplication rule for the intersection of two events.

$$P(A \cap B) = P(A)\,P(B \mid A) = P(B)\,P(A \mid B)$$

When are two events independent, and what does the multiplication rule become?

Events $A$ and $B$ are independent if the occurrence of one does not affect the other: $P(A \mid B) = P(A)$. Then $P(A \cap B) = P(A)\,P(B)$.

What is the difference between mutually exclusive events and independent events?

Mutually exclusive events cannot occur together ($P(A\cap B)=0$). Independent events can occur together but one does not affect the other's probability. Non-trivial mutually exclusive events are dependent.

State Bayes' theorem.

$$P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}$$

What is the formula for permutations of $n$ objects taken $r$ at a time?

$$_{n}P_{r} = \frac{n!}{(n-r)!}$$ used when order matters.

What is the formula for combinations of $n$ objects taken $r$ at a time?

$$_{n}C_{r} = \binom{n}{r} = \frac{n!}{r!\,(n-r)!}$$ used when order does not matter.

When do you use a permutation versus a combination?

Use a permutation when the order of selection matters (arrangements). Use a combination when order does not matter (selections/groups).

Define a random variable.

A random variable is a function that assigns a numerical value to each outcome of a random experiment. It may be discrete or continuous.

How is the expected value (mean) of a discrete random variable computed?

$$E(X) = \mu = \sum_{i} x_{i}\,P(x_{i})$$ the probability-weighted sum of its values.

How is the variance of a discrete random variable computed?

$$\operatorname{Var}(X) = E[(X-\mu)^{2}] = \sum_{i}(x_{i}-\mu)^{2}P(x_{i}) = E(X^{2}) - [E(X)]^{2}$$

What conditions define a binomial experiment?

A fixed number $n$ of independent trials, each with two outcomes (success/failure), and a constant probability of success $p$ on every trial.

State the binomial probability formula.

$$P(X=k) = \binom{n}{k} p^{k}(1-p)^{n-k}$$ for $k = 0,1,\dots,n$.

What are the mean and standard deviation of a binomial distribution?

$\mu = np$ and $\sigma = \sqrt{np(1-p)}$.

What does a Poisson distribution model, and what is its probability formula?

It models the number of events in a fixed interval given a constant average rate $\lambda$. $$P(X=k) = \frac{\lambda^{k} e^{-\lambda}}{k!}$$ with mean and variance both equal to $\lambda$.

What are the key properties of the normal distribution curve?

It is bell-shaped, symmetric about the mean $\mu$, has mean = median = mode, and the total area under the curve equals $1$.

What is the standard normal distribution?

A normal distribution with mean $\mu = 0$ and standard deviation $\sigma = 1$, denoted $N(0,1)$, obtained by converting values to z-scores.

State the Central Limit Theorem.

For sufficiently large sample size $n$, the sampling distribution of the sample mean $\bar{x}$ is approximately normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$, regardless of the population's shape.

What is the standard error of the mean?

The standard deviation of the sampling distribution of the sample mean: $$SE = \frac{\sigma}{\sqrt{n}}$$

What is a confidence interval for a population mean (known $\sigma$)?

$$\bar{x} \pm z^{*}\frac{\sigma}{\sqrt{n}}$$ where $z^{*}$ is the critical value for the chosen confidence level (e.g. $1.96$ for 95%).

Distinguish a null hypothesis $H_{0}$ from an alternative hypothesis $H_{a}$.

$H_{0}$ states no effect or no difference (status quo). $H_{a}$ states the effect or difference the researcher is testing for. The test seeks evidence against $H_{0}$.

Define Type I and Type II errors.

A Type I error rejects a true $H_{0}$ (false positive), with probability $\alpha$. A Type II error fails to reject a false $H_{0}$ (false negative), with probability $\beta$.

What is a p-value, and how is it used to decide significance?

The p-value is the probability of observing results at least as extreme as the data, assuming $H_{0}$ is true. Reject $H_{0}$ if p-value $\leq \alpha$.

What does the correlation coefficient $r$ measure, and what is its range?

$r$ measures the strength and direction of a linear relationship between two variables, with $-1 \leq r \leq 1$. Values near $\pm 1$ indicate strong linear association; near $0$ indicates weak/no linear association.

Why does correlation not imply causation?

A strong correlation only indicates association; a lurking/confounding variable or coincidence may cause both, so one variable does not necessarily cause changes in the other.

What is the equation of a least-squares regression line?

$$\hat{y} = b_{0} + b_{1}x$$ where $b_{1} = r\frac{s_{y}}{s_{x}}$ is the slope and $b_{0} = \bar{y} - b_{1}\bar{x}$ is the intercept.

Compare a probability sample with a non-probability (convenience) sample.

In a probability sample every unit has a known nonzero chance of selection (e.g. simple random sampling), allowing valid inference. A convenience sample selects easily available units and is prone to bias.

Define the expected value and explain its meaning in a fair game.

The expected value $E(X)=\sum x_{i}P(x_{i})$ is the long-run average outcome. A game is fair when the expected net gain is $0$.