STAT101 - Introduction to Probability & Statistics Flashcards

Question 1

Mean

Accepted Answer

The arithmetic average of a dataset, calculated by taking the sum of all values divided by the total number of observations. Population mean: $\mu = \frac{\sum x}{N}$; Sample mean: $\bar{x} = \frac{\sum x}{n}$.

Question 2

Median

Accepted Answer

The middle value in a dataset when the observations are ordered from smallest to largest. If the dataset has an even number of observations, it is the average of the two middle numbers. Unlike the mean, it is resistant to outliers.

Question 3

Mode

Accepted Answer

The value that appears most frequently in a dataset. A dataset can have one mode (unimodal), multiple modes (bimodal/multimodal), or no mode at all if all values occur with equal frequency.

Question 4

Variance

Accepted Answer

A measure of how far a set of numbers is spread out from their average. Sample variance ($s^2$) uses $n-1$ degrees of freedom: $s^2 = \frac{\sum (x - \bar{x})^2}{n - 1}$. Population variance uses $N$: $\sigma^2 = \frac{\sum (x - \mu)^2}{N}$.

Question 5

Standard Deviation

Accepted Answer

The square root of the variance, representing the average distance of data points from the mean in the same units as the original data. Sample: $s = \sqrt{s^2}$; Population: $\sigma = \sqrt{\sigma^2}$.

Question 6

Interquartile Range (IQR)

Accepted Answer

A measure of statistical dispersion representing the spread of the middle 50% of the data, calculated as $IQR = Q_3 - Q_1$. It is resistant to extreme values and outliers.

Question 7

Kolmogorov's Axioms of Probability

Accepted Answer

The three foundational rules of probability theory:
1. Non-negativity: $P(A) \ge 0$ for any event $A$.
2. Normalization: $P(\Omega) = 1$, where $\Omega$ is the sample space.
3. Additivity: For mutually exclusive events $A_1, A_2, \dots$, $P(\bigcup A_i) = \sum P(A_i)$.

Question 8

Conditional Probability

Accepted Answer

The probability of an event $A$ occurring given that event $B$ has already occurred, defined as $P(A|B) = \frac{P(A \cap B)}{P(B)}$ provided $P(B) > 0$.

Question 9

Bayes' Theorem

Accepted Answer

A mathematical formula used to update the probability of a hypothesis as more evidence becomes available:
$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$
where $P(A)$ is the prior, $P(A|B)$ is the posterior, and $P(B|A)$ is the likelihood.

Question 10

Binomial Distribution

Accepted Answer

A discrete probability distribution modeling the number of successes $k$ in $n$ independent yes/no trials, each with success probability $p$:
$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

Question 11

Poisson Distribution

Accepted Answer

A discrete probability distribution expressing the probability of a given number of events $k$ occurring in a fixed interval of time or space, given a constant rate $\lambda$:
$$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$$

Question 12

Normal Distribution

Accepted Answer

A continuous, symmetric, bell-shaped probability distribution defined by its mean $\mu$ and variance $\sigma^2$, denoted $X \sim \mathcal{N}(\mu, \sigma^2)$. Roughly 68%, 95%, and 99.7% of values lie within 1, 2, and 3 standard deviations of the mean.

Question 13

Standard Normal Distribution (Z-Score)

Accepted Answer

A normal distribution with $\mu = 0$ and $\sigma = 1$. Any normal random variable $X$ can be transformed into a standard normal variable $Z$ using $Z = \frac{X - \mu}{\sigma}$.

Question 14

Central Limit Theorem (CLT)

Accepted Answer

States that the sampling distribution of the sample mean ($\bar{x}$) approaches a normal distribution as the sample size $n$ becomes large (typically $n \ge 30$), regardless of the shape of the underlying population distribution.

Question 15

Null Hypothesis ($H_0$)

Accepted Answer

The default statement being tested in a statistical hypothesis test, usually asserting no effect, no difference, or no relationship between variables (e.g., $H_0: \mu_1 = \mu_2$).

Question 16

Alternative Hypothesis ($H_1$ or $H_a$)

Accepted Answer

The hypothesis that contradicts $H_0$, representing the claim or effect that the researcher aims to gather evidence to support (e.g., $H_a: \mu_1 
eq \mu_2$).

Question 17

p-value

Accepted Answer

The probability of obtaining a test statistic at least as extreme as the observed result, assuming the null hypothesis ($H_0$) is true. If $p \le \alpha$ (significance level), we reject $H_0$.

Question 18

Type I Error vs. Type II Error

Accepted Answer

- **Type I Error (False Positive)**: Rejecting $H_0$ when it is actually true ($
P(	ext{Type I}) = \alpha$).
- **Type II Error (False Negative)**: Failing to reject $H_0$ when it is actually false ($
P(	ext{Type II}) = \beta$).

Question 19

Confidence Interval (CI)

Accepted Answer

A range of plausible values for an unknown population parameter calculated from sample data at a chosen confidence level ($1 - \alpha$). For a population mean with known $\sigma$: $\bar{x} \pm Z^* \frac{\sigma}{\sqrt{n}}$.

Question 20

Pearson Correlation Coefficient ($r$)

Accepted Answer

A measure of the strength and direction of a linear relationship between two quantitative variables, ranging from $-1$ (perfect negative) to $+1$ (perfect positive): $r = \frac{\sum(x - \bar{x})(y - \bar{y})}{\sqrt{\sum(x - \bar{x})^2 \sum(y - \bar{y})^2}}$.

STAT101 - Introduction to Probability & Statistics

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