Statistics & Probability Probability Distributions Syllabus

State the mean and variance of an Exponential distribution with rate $\lambda$.

$E[X]=\dfrac{1}{\lambda}$ and $\operatorname{Var}(X)=\dfrac{1}{\lambda^{2}}$.

Write the cdf of an Exponential$(\lambda)$ distribution and state its memoryless property.

$F(x)=1-e^{-\lambda x}$ for $x\ge 0$. Memoryless: $P(X>s+t\mid X>s)=P(X>t)$.

What does the Gamma distribution model, and what are its shape and rate parameters?

It models the waiting time until the $k$-th event in a Poisson process. Shape $\alpha$ (or $k$) and rate $\beta$ (or scale $1/\beta$). pdf: $f(x)=\dfrac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}$, $x\ge 0$.

State the mean and variance of a Gamma distribution with shape $\alpha$ and rate $\beta$.

$E[X]=\dfrac{\alpha}{\beta}$ and $\operatorname{Var}(X)=\dfrac{\alpha}{\beta^{2}}$.

What is the Beta distribution used for, and what is its support?

The Beta$(\alpha,\beta)$ distribution models a random variable on $[0,1]$, often a probability or proportion. Support $x\in[0,1]$, pdf $f(x)=\dfrac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$.

State the mean of a Beta$(\alpha,\beta)$ distribution.

$E[X]=\dfrac{\alpha}{\alpha+\beta}$.

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.

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

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

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.

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

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

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.

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

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

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.

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

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

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 $P(Z<-z)=1-\Phi(z)=P(Z>z)$.

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 $\mu\pm 3\sigma$.

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.

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.

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.

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).

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 $\dfrac{\sigma}{\sqrt{n}}$.

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.

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.

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.

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

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

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 $N\!\left(p,\sqrt{\dfrac{p(1-p)}{n}}\right)$.

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 $t=\dfrac{\bar{X}-\mu}{s/\sqrt{n}}$.

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, $df=n-1$.

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.

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.