Statistics & Probability Probability Distributions Flashcards

24 sample cards from the Probability Distributions deck

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

1. What does the Weibull distribution model, and what does its shape parameter control?

   It models time-to-failure/reliability and lifetimes. Shape $k$ controls the failure rate: $k<1$ decreasing hazard, $k=1$ constant hazard (reduces to Exponential), $k>1$ increasing hazard.

2. Write the pdf of a Weibull distribution with shape $k$ and scale $\lambda$.

   $$f(x)=\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}},\quad x\ge 0.$$

3. List the key shape properties of the Normal curve.

   It is bell-shaped, symmetric about the mean $\mu$, unimodal, with mean = median = mode, and asymptotic tails that approach but never touch the horizontal axis. Total area under the curve = 1.

4. Write the pdf of a Normal distribution with mean $\mu$ and variance $\sigma^{2}$.

   $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}.$$

5. Where are the inflection points of a Normal curve located?

   At $x=\mu-\sigma$ and $x=\mu+\sigma$, i.e. one standard deviation from the mean on each side.

6. What are the mean and standard deviation of the standard Normal distribution, and its notation?

   Mean $\mu=0$ and standard deviation $\sigma=1$, denoted $Z\sim N(0,1)$.

7. How do you convert a Normal value $x$ into a $z$-score (standardize it)?

   $$z=\frac{x-\mu}{\sigma}.$$ The $z$-score gives the number of standard deviations $x$ is from the mean.

8. What quantity does a standard Normal ($z$) table give?

   It gives the cumulative area (probability) to the left of a given $z$-value: $P(Z\le z)=\Phi(z)$.

9. Using symmetry, how do you find $P(Z > z)$ and $P(Z < -z)$ from a $z$-table?

   $P(Z>z)=1-\Phi(z)$, and by symmetry $P(Z<-z)=1-\Phi(z)=P(Z>z)$.

10. State the Empirical Rule (68-95-99.7) for a Normal distribution.

    Approximately $68\%$ of data lies within $\mu\pm\sigma$, $95\%$ within $\mu\pm 2\sigma$, and $99.7\%$ within $\mu\pm 3\sigma$.

11. By the Empirical Rule, what percentage of data lies beyond $\mu\pm 2\sigma$?

    About $5\%$ total lies outside $\mu\pm 2\sigma$, so roughly $2.5\%$ in each tail.

12. What graphical tool is used to assess Normality, and what pattern indicates a Normal distribution?

    A normal probability plot (Q-Q plot). If the data are approximately Normal, the points fall roughly along a straight line.

13. Besides a Q-Q plot, name features to check when assessing Normality of data.

    Check for approximate symmetry (skewness near 0), a single mode (unimodal bell shape), no strong outliers, and kurtosis near that of a Normal. Formal tests include Shapiro-Wilk and Anderson-Darling.

14. If a population has mean $\mu$ and standard deviation $\sigma$, what are the mean and standard deviation of the sample mean $\bar{X}$ for samples of size $n$?

    $E[\bar{X}]=\mu$ and $\operatorname{SD}(\bar{X})=\dfrac{\sigma}{\sqrt{n}}$ (the standard error of the mean).

15. State the Central Limit Theorem.

    For a sufficiently large sample size $n$, the distribution of the sample mean $\bar{X}$ is approximately Normal, regardless of the population's shape, with mean $\mu$ and standard deviation $\dfrac{\sigma}{\sqrt{n}}$.

16. What sample size is commonly cited as "large enough" for the Central Limit Theorem to apply?

    Commonly $n\ge 30$ is used as a rule of thumb; fewer may suffice if the population is already roughly symmetric/Normal.

17. Define the standard error of the mean and explain what it measures.

    $\operatorname{SE}(\bar{X})=\dfrac{\sigma}{\sqrt{n}}$ (or $\dfrac{s}{\sqrt{n}}$ when $\sigma$ is unknown). It measures the variability of the sample mean from sample to sample.

18. How does increasing the sample size $n$ affect the standard error?

    It decreases the standard error, since $\operatorname{SE}=\dfrac{\sigma}{\sqrt{n}}$; quadrupling $n$ halves the standard error.

19. For a sample proportion $\hat{p}$, what are its mean and standard deviation (standard error)?

    $E[\hat{p}]=p$ and $\operatorname{SD}(\hat{p})=\sqrt{\dfrac{p(1-p)}{n}}$.

20. What conditions make the sampling distribution of $\hat{p}$ approximately Normal?

    When $np\ge 10$ and $n(1-p)\ge 10$ (and observations are independent), $\hat{p}$ is approximately $N\!\left(p,\sqrt{\dfrac{p(1-p)}{n}}\right)$.

21. When is the Student's $t$-distribution used instead of the standard Normal?

    When estimating a mean with the population standard deviation $\sigma$ unknown (using sample $s$), especially for small samples. The test statistic is $t=\dfrac{\bar{X}-\mu}{s/\sqrt{n}}$.

22. How does the $t$-distribution's shape compare to the standard Normal, and what parameter governs it?

    The $t$-distribution is symmetric and bell-shaped but has heavier (fatter) tails than the Normal. Its shape is governed by the degrees of freedom; for a one-sample mean, $df=n-1$.

23. What happens to the $t$-distribution as the degrees of freedom increase?

    As $df\to\infty$, the $t$-distribution approaches the standard Normal $N(0,1)$; the tails become thinner.

24. Compare discrete distributions (Bernoulli, Binomial, Poisson, Geometric, Hypergeometric) by what they count.

    Bernoulli: success in one trial. Binomial: successes in $n$ independent trials (with replacement). Geometric: trials until first success. Negative Binomial: trials until $r$-th success. Hypergeometric: successes in $n$ draws without replacement. Poisson: events in a fixed interval.

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Probability Distributions flashcards FAQ