6 Common Probability Models Free Online FlashCards

How do discrete and continuous random variables differ?

A discrete variable has a finite or countably infinite set of values and uses a PMF; a continuous variable can take any value in an interval and uses a PDF.

What does a cumulative distribution function measure?

The cumulative distribution function is $F_X(x)=P(X\le x)$. It gives the probability that $X$ is at most $x$.

What four conditions define a binomial model?

The binomial model requires a fixed number of trials, two outcomes per trial, independence, and the same success probability on every trial.

What is the binomial probability of exactly $k$ successes?

For $X\sim\operatorname{Binomial}(n,p)$, $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$.

What does the geometric distribution count?

The geometric distribution counts the number of independent Bernoulli trials required to obtain the first success.

What does the negative binomial distribution count?

The negative binomial distribution counts the trial number on which the $r$th success occurs, extending the geometric model beyond the first success.

When is the hypergeometric model appropriate?

The hypergeometric model applies to sampling without replacement from a finite population, where draws are generally dependent.

What are the mean and variance of a Poisson variable?

For $X\sim\operatorname{Poisson}(\lambda)$, $E(X)=\lambda$ and $\operatorname{Var}(X)=\lambda$.

What characterizes a uniform distribution on $[a,b]$?

For $X\sim\operatorname{Uniform}(a,b)$, every equal-length subinterval within $[a,b]$ has the same probability.

What is the exponential survival probability?

For an exponential variable with rate $\lambda$, $P(X>x)=e^{-\lambda x}$.

What are the defining shape features of a normal distribution?

A normal variable is symmetric and bell-shaped; its mean, median, and mode are equal, with center $\mu$ and spread $\sigma$.

How is a normal observation standardized?

A z-score is $Z=\frac{X-\mu}{\sigma}$; it expresses a value's distance from the mean in standard deviations.