AP Statistics 2.5-2.6: Random Variables and Probability Distributions

Discrete probability distribution function (PDF)

A table or rule that assigns a probability $P(X=x)$ to each possible value of a discrete random variable $X$. Every probability must satisfy $0 \le P(X=x) \le 1$, and the probabilities over all possible values must sum to $1$.

How can you check whether a table of values and probabilities is a valid discrete PDF?

Verify that every listed probability is in the interval $[0,1]$ and calculate the total probability. The table is valid only if the total is exactly $1$, allowing for reasonable rounding.

Random variable $X$

A variable whose numerical value is determined by the outcome of a random process. In a discrete setting, $X$ can take a countable set of separate values, such as the number of particles, occurrences, or successes.

How should $P(X=x)$ be interpreted?

$P(X=x)$ is the probability that the random variable $X$ takes the specific value $x$.

How do you construct a discrete PDF from observed counts?

For each possible outcome $x$, divide its observed count by the total number of observations: $P(X=x)=\frac{\text{count for }x}{\text{total count}}$. The resulting relative frequencies should sum to $1$.

A study records the number of times a patient rings a nurse during a shift. Why is this variable discrete?

It is a count, so its possible values are separate whole numbers such as $0,1,2,3,\ldots$, rather than every value in an interval.

How is a probability distribution table typically organized for a discrete random variable?

One column lists each possible value of $X$, denoted $x$, and a second column lists its probability, $P(X=x)$.

If a discrete PDF lists $P(X=0)=0.01$, $P(X=1)=0.04$, $P(X=2)=0.15$, and $P(X=3)=0.80$, what is $P(X\le 2)$?

Add the probabilities for the included values: $P(X\le2)=0.01+0.04+0.15=0.20$.

How are probabilities for mutually exclusive values combined in a discrete distribution?

Add them. For example, $P(a\le X\le b)=\sum_{x=a}^{b}P(X=x)$ when the possible values are discrete and the events do not overlap.

Complement rule for a discrete random variable

The probability of an event and its complement sum to $1$: $P(X\text{ does not satisfy }A)=1-P(A)$. For example, $P(X\ge1)=1-P(X=0)$.

Bernoulli trial

A single random experiment with exactly two possible outcomes, conventionally called success and failure. If the success probability is $p$, the failure probability is $q=1-p$.

Binomial random variable

A variable that counts the number of successes in a fixed number $n$ of Bernoulli trials, where the trials are independent and each trial has the same success probability $p$. It is written $X\sim\operatorname{Binomial}(n,p)$.

What four conditions must be met for a binomial model?

There must be a fixed number $n$ of trials, each trial must have two outcomes, the trials must be independent, and the probability of success $p$ must remain constant from trial to trial.

What do the parameters $n$, $p$, and $q$ represent in a binomial distribution?

$n$ is the number of trials, $p$ is the probability of success on each trial, and $q=1-p$ is the probability of failure on each trial.

What values can a binomial random variable $X$ take?

If $X$ counts successes in $n$ trials, then $X$ can take the integer values $0,1,2,\ldots,n$.

Binomial probability mass function

For $X\sim\operatorname{Binomial}(n,p)$, the probability of exactly $k$ successes is $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}=\binom{n}{k}p^kq^{n-k}$.

Why does the binomial formula include the factor $\binom{n}{k}$?

The factor $p^kq^{n-k}$ gives the probability of one particular sequence containing $k$ successes and $n-k$ failures. The binomial coefficient $\binom{n}{k}=\frac{n!}{k!(n-k)!}$ counts the different sequences in which the $k$ successes can occur.

How do you calculate the probability of exactly 4 successes in 6 independent trials when $p=0.30$?

Use $P(X=4)=\binom{6}{4}(0.30)^4(0.70)^2$. This evaluates to approximately $0.0595$.

How is the probability of at most $k$ successes represented for a binomial random variable?

$P(X\le k)=\sum_{i=0}^{k}\binom{n}{i}p^i(1-p)^{n-i}$. It is found by adding the probabilities of $0,1,\ldots,k$ successes.

How can you calculate a binomial upper-tail probability such as $P(X\ge k)$ efficiently?

Use the complement when convenient: $P(X\ge k)=1-P(X\le k-1)$. This often requires fewer individual binomial probabilities.

Expected value of a binomial random variable

If $X\sim\operatorname{Binomial}(n,p)$, then $E(X)=np$. It represents the long-run average number of successes, not necessarily a value that must occur in one set of trials.

Variance and standard deviation of a binomial random variable

For $X\sim\operatorname{Binomial}(n,p)$, $\operatorname{Var}(X)=np(1-p)=npq$, and the standard deviation is $\sigma_X=\sqrt{np(1-p)}$.

Why is the expected number of binomial successes equal to $np$?

The total number of successes is the sum of $n$ indicator variables, each with expected value $p$. By linearity of expectation, $E(X)=p+p+\cdots+p=np$.

What happens to binomial variability when the success probability approaches $0$ or $1$?

The variance $np(1-p)$ becomes smaller because outcomes become more predictable. For $p=0$ or $p=1$, the count is fixed and the variance is $0$.

How does a Bernoulli distribution relate to a binomial distribution?

A Bernoulli distribution is the special case of a binomial distribution with $n=1$. It describes one success/failure trial, whereas a binomial distribution counts successes across multiple trials.

What is the difference between a discrete PDF and a binomial PDF?

A discrete PDF is any valid probability assignment for a discrete random variable. A binomial PDF is a specific discrete PDF generated by counting successes under the binomial conditions and using $\binom{n}{k}p^k(1-p)^{n-k}$.

A binomial experiment has $n=20$ and $p=0.25$. What are its mean and standard deviation?

The mean is $E(X)=np=20(0.25)=5$. The standard deviation is $\sqrt{np(1-p)}=\sqrt{20(0.25)(0.75)}\approx1.94$.