3 Probability Distributions Free Online FlashCards

What does a probability distribution describe?

A probability distribution describes the possible values of a random variable and the probability associated with each value or range of values.

What is a discrete random variable?

A discrete random variable takes countable values, such as $0, 1, 2, \ldots$.

What is a continuous random variable?

A continuous random variable can take any value within a range, as measurements such as time or weight can.

What does the cumulative distribution function $F(x)$ give?

The cumulative distribution function is $F(x)=P(X\le x)$; it gives the probability that the variable is at or below $x$.

What conditions make a binomial model appropriate?

Use a binomial model for successes in a fixed number of independent trials, each with two mutually exclusive outcomes and the same success probability $p$.

What characterizes a normal distribution?

A normal distribution is a continuous, symmetric, bell-shaped model described by its mean $\mu$ and standard deviation $\sigma$; the area under its curve is $1$.

What does a probability mass function (PMF) give?

A probability mass function gives the probability of each discrete value, $P(X=k)$; its probabilities across all possible values sum to $1$.

How does a probability density function (PDF) represent probability?

A probability density function describes how probability is distributed; the probability of an interval is the area under the curve over that interval.

What is the probability of one exact value of a continuous random variable?

For a continuous random variable, the probability of any one exact value is $0$.

What is the binomial probability of exactly $k$ successes in $n$ trials?

For $k=0,1,\ldots,n$, the binomial probability is $P(X=k)=C(n,k)p^k(1-p)^{n-k}$.

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

For a binomial variable with $n$ trials and success probability $p$, the mean is $np$ and the standard deviation is $\sqrt{np(1-p)}$.

What does $C(n,k)$ count in the binomial formula?

The coefficient $C(n,k)$ counts the different ways to obtain $k$ successes in $n$ trials.