Random Variables and the Binomial Distribution: Quick Revision

A brief note on Random Variables and the Binomial Distribution. Read it before a test. Try each question at the end before you open its answer.

Random variables

• Distribution: a table of values $x$ and probabilities $P(X = x)$ that add up to $1$.
• Mean: $E(X) = \sum x\,P(X = x)$.
• Variance: $\text{Var}(X) = E(X^2) - [E(X)]^2$, and the standard deviation is its square root.
• Linear change: $E(aX + b) = aE(X) + b$ and $\text{Var}(aX + b) = a^2\,\text{Var}(X)$.
• Fair game: the expected net gain is $0$.

The binomial distribution

• Conditions: a fixed number $n$ of trials, two outcomes, the same $p$, independent trials.
• Formula: $P(X = k) = \binom{n}{k}p^k q^{n-k}$, written $X \sim B(n, p)$.
• Mean and variance: $np$ and $npq$.
• At least one: $1 - q^n$, and $n \ge \dfrac{\ln(1 - t)}{\ln q}$ for a target $t$, rounded up.
• Most likely value: the integer part of $(n + 1)p$. It is two values when $(n + 1)p$ is a whole number.

The normal approximation

• When: $np$ and $nq$ both large, at least $5$ in A-Level texts and $10$ in AP.
• Model: $X \approx N(np, npq)$.
• Continuity correction: $P(X \le k) \approx \Phi\!\left(\dfrac{k + 0.5 - np}{\sqrt{npq}}\right)$.
• Not required in the Indian syllabus. It is used in A-Level and AP.

A worked example

A fair coin is tossed $4$ times. Let $X$ be the number of heads.

1. Exactly two. $P(X = 2) = \binom{4}{2}\left(\dfrac{1}{2}\right)^4 = \dfrac{6}{16} = \dfrac{3}{8}$.
2. Mean and variance. $np = 2$ and $npq = 4 \times \dfrac{1}{2} \times \dfrac{1}{2} = 1$.
3. At least three. $P(X = 3) + P(X = 4) = \dfrac{4}{16} + \dfrac{1}{16} = \dfrac{5}{16}$.

Practice with new numbers

A program to try

The program finds the three answers of the worked example with exact fractions.

Mistakes to avoid

• Leaving out $\binom{n}{k}$. The product $p^k q^{n-k}$ is for one order only.
• Using the binomial without independence. Draws without replacement from a small pack are not binomial.
• Adding $b$ to the variance. Only the factor $a$ matters, and it is squared.
• Forgetting to round up the number of trials needed for a target chance.

Check yourself

1. A random variable $X$ has $P(X = 0) = 0.4$, $P(X = 1) = 0.4$ and $P(X = 2) = 0.2$. Find the mean and the variance.

2. $X \sim B(4, 0.3)$. Find $P(X = 2)$.

3. A binomial variable has mean $2$ and variance $1.6$. Find $n$ and $p$.

4. A fair die is thrown $5$ times. Find the probability of at least one six.

5. Is the number of throws of a die needed to get the first six a binomial variable?

Sources

• OpenStax, Introductory Statistics, Chapter 4, sections on discrete random variables, the binomial distribution and the geometric distribution.
• NCERT, Mathematics Part II for Class XII, Chapter 13, sections on random variables, mean and variance, and the binomial distribution.