Mathematics Statistics and Probability Flashcards

24 sample cards from the Statistics and Probability deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. State Bayes' theorem.

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

2. 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.

3. 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.

4. 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).

5. 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.

6. 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.

7. How is the variance of a discrete random variable computed?

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

8. 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.

9. State the binomial probability formula.

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

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

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

11. 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 $\lambda$.

12. 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 $1$.

13. 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.

14. 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.

15. What is the standard error of the mean?

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

16. 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%).

17. 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 $H_{0}$.

18. 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 $\beta$.

19. 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 $\leq \alpha$.

20. 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.

21. 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.

22. 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.

23. 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.

24. 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 $0$.

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