Discrete random variables: probability distributions, expectation, variance

Discrete Random Variables – Cambridge AS & A‑Level Mathematics (9709) – P&S 1

1. Basic Probability Rules (Prerequisite)

• Complement: \(P(A^{c}) = 1 - P(A)\).
• Addition (mutually exclusive events): \(P(A\cup B)=P(A)+P(B)\).
• Addition (general): \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\).
• Multiplication (independent events): \(P(A\cap B)=P(A)\,P(B)\) when \(A\) and \(B\) are independent.
• Conditional probability: \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\) (provided \(P(B)>0\)).
• Independence test: \(A\) and \(B\) are independent iff \(P(A\mid B)=P(A)\) (equivalently \(P(A\cap B)=P(A)P(B)\)).
• Tree/diagram method: useful for sequential experiments (e.g., rolling dice, drawing cards). It visualises both the multiplication rule and conditional probabilities.

2. Representation of Data (Syllabus 5.1)

Choosing the right display helps to see patterns, outliers and the shape of a data set.

• Stem‑and‑leaf plot: retains the original values while showing distribution. Example for the data \(\{12,13,14,15,21,22,23,31\}\):

  1 | 2 3 4 5
  2 | 1 2 3
  3 | 1
          

• Box‑plot (five‑number summary): minimum, \(Q_1\), median, \(Q_3\), maximum. Shows centre, spread and possible outliers.
• Histogram: groups data into class intervals; the height of each bar is proportional to the frequency (or relative frequency) of the interval.
• Measures of central tendency & spread: mean \(\bar{x}\), median, mode, range, inter‑quartile range (IQR), variance and standard deviation.

3. Counting Techniques – Permutations & Combinations (Syllabus 5.2)

These tools let us count the number of equally likely outcomes that a probability problem requires.

• Fundamental counting principle: if an experiment can be performed in \(n_1\) ways, then \(n_2\) ways, …, \(n_k\) ways, the total number of outcomes is \(n_1\times n_2\times\cdots\times n_k\).
• Permutations (ordered selections): \[ P(n,r)=\frac{n!}{(n-r)!} \] Example: “How many 3‑digit numbers can be formed from the digits \(\{1,2,3,4,5\}\) without repetition?” \(P(5,3)=\frac{5!}{2!}=60\) numbers.
• Combinations (unordered selections): \[ \binom{n}{r}= \frac{n!}{r!\,(n-r)!} \] Example: “From a class of 12 students, in how many ways can a committee of 4 be chosen?” \(\displaystyle\binom{12}{4}=495\) ways.

4. Discrete Random Variables (Syllabus 5.3)

4.1 Definition

A discrete random variable \(X\) can take only a finite or countably infinite set of values \(\{x_1,x_2,\dots\}\). The possible outcomes must be exhaustive (cover all cases) and mutually exclusive (no two can occur simultaneously). For each value \(x_i\)

• \(0\le P(X=x_i)\le 1\),
• \(\displaystyle\sum_{i}P(X=x_i)=1\).

4.2 Probability Mass Function (PMF)

The function

\[ p_X(x)=P(X=x) \]

is called the probability mass function. It completely specifies the distribution of a discrete random variable.

4.3 Cumulative Distribution Function (CDF)

The cumulative distribution function is

\[ F_X(x)=P(X\le x)=\sum_{t\le x}p_X(t). \]

Key properties:

• \(0\le F_X(x)\le 1\) for all real \(x\);
• non‑decreasing in \(x\);
• \(\displaystyle\lim_{x\to-\infty}F_X(x)=0,\qquad \lim_{x\to\infty}F_X(x)=1.\)

4.4 Example – Number of Heads in Three Coin Tosses

Let \(X\) be the number of heads when a fair coin is tossed three times.

\(x\)	Ordered outcomes	# outcomes	\(p_X(x)\)
0	TTT	1	\(\frac{1}{8}\)
1	HTT, THT, TTH	3	\(\frac{3}{8}\)
2	HHT, HTH, THH	3	\(\frac{3}{8}\)
3	HHH	1	\(\frac{1}{8}\)

From the table we obtain the CDF, e.g. \(F_X(1)=\frac{1}{8}+\frac{3}{8}= \frac{1}{2}\).

5. Expectation and Variance (Syllabus 5.4)

5.1 Expectation (Mean)

The expectation of \(X\) is the weighted average of its possible values:

\[ E(X)=\mu=\sum_{i}x_i\,p_X(x_i). \]

Linearity (valid for any constants \(a,b\)):

\[ E(aX+b)=aE(X)+b. \]

5.2 Variance and Standard Deviation

The variance measures the average squared deviation from the mean:

\[ \operatorname{Var}(X)=\sigma^{2}=E\!\big[(X-\mu)^{2}\big]=\sum_{i}(x_i-\mu)^{2}p_X(x_i). \]

Computational shortcut (often easier):

\[ \operatorname{Var}(X)=E(X^{2})-\big[E(X)\big]^{2}. \]

Standard deviation: \(\displaystyle \sigma=\sqrt{\operatorname{Var}(X)}.\)

Scaling rule:

\[ \operatorname{Var}(aX+b)=a^{2}\operatorname{Var}(X). \]

6. Common Discrete Distributions (P&S 1)

6.1 Binomial Distribution

Number of successes in \(n\) independent Bernoulli trials, each with success probability \(p\).

\[ X\sim\operatorname{Bin}(n,p),\qquad p_X(k)=\binom{n}{k}p^{k}(1-p)^{\,n-k},\;k=0,1,\dots,n. \]

Mean and variance:

\[ E(X)=np,\qquad \operatorname{Var}(X)=np(1-p). \]

6.2 Geometric Distribution

Number of trials required to obtain the first success.

\[ X\sim\operatorname{Geom}(p),\qquad p_X(k)=(1-p)^{k-1}p,\;k=1,2,\dots \]

Mean and variance:

\[ E(X)=\frac{1}{p},\qquad \operatorname{Var}(X)=\frac{1-p}{p^{2}}. \]

6.3 Poisson Distribution – Preview (P&S 2)

Useful for modelling the number of events occurring in a fixed interval when events are rare.

\[ X\sim\operatorname{Pois}(\lambda),\qquad p_X(k)=\frac{e^{-\lambda}\lambda^{k}}{k!},\;k=0,1,2,\dots \]

Mean and variance are both \(\lambda\). It can be derived as the limit of a binomial distribution with \(n\to\infty,\;p\to0\) while \(np=\lambda\) stays constant.

7. Worked Example – Sum of Two Dice

7.1 Problem statement

A fair six‑sided die is rolled twice. Let \(X\) be the sum of the two results. Determine the PMF, the mean \(E(X)\), the variance \(\operatorname{Var}(X)\) and give the CDF.

7.2 Counting the outcomes (using the product rule)

Each roll has 6 equally likely results, so the ordered pair \((a,b)\) has \(6^{2}=36\) equally likely possibilities.

Sum \(x\)	Ordered pairs \((a,b)\)	# outcomes	\(p_X(x)\)
2	(1,1)	1	\(\frac{1}{36}\)
3	(1,2),(2,1)	2	\(\frac{2}{36}\)
4	(1,3),(2,2),(3,1)	3	\(\frac{3}{36}\)
5	(1,4),(2,3),(3,2),(4,1)	4	\(\frac{4}{36}\)
6	(1,5),(2,4),(3,3),(4,2),(5,1)	5	\(\frac{5}{36}\)
7	(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)	6	\(\frac{6}{36}\)
8	(2,6),(3,5),(4,4),(5,3),(6,2)	5	\(\frac{5}{36}\)
9	(3,6),(4,5),(5,4),(6,3)	4	\(\frac{4}{36}\)
10	(4,6),(5,5),(6,4)	3	\(\frac{3}{36}\)
11	(5,6),(6,5)	2	\(\frac{2}{36}\)
12	(6,6)	1	\(\frac{1}{36}\)

7.3 Expectation

\[ E(X)=\sum_{x=2}^{12} x\,p_X(x) =\frac{1}{36}\bigl(2\cdot1+3\cdot2+4\cdot3+5\cdot4+6\cdot5+7\cdot6 +8\cdot5+9\cdot4+10\cdot3+11\cdot2+12\cdot1\bigr)=7. \]

7.4 Second moment and variance

\[ E(X^{2})=\sum_{x=2}^{12} x^{2}\,p_X(x) =\frac{1}{36}\bigl(2^{2}\cdot1+3^{2}\cdot2+4^{2}\cdot3+\dots+12^{2}\cdot1\bigr) =\frac{91}{6}. \] \[ \operatorname{Var}(X)=E(X^{2})-[E(X)]^{2} =\frac{91}{6}-7^{2} =\frac{35}{12}\approx2.92. \]

7.5 Cumulative distribution function

\[ F_X(x)=P(X\le x)= \begin{cases} 0, & x<2,\\[4pt] \frac{1}{36}, & 2\le x<3,\\[4pt] \frac{3}{36}, & 3\le x<4,\\[4pt] \frac{6}{36}, & 4\le x<5,\\[4pt] \frac{10}{36}, & 5\le x<6,\\[4pt] \frac{15}{36}, & 6\le x<7,\\[4pt] \frac{21}{36}, & 7\le x<8,\\[4pt] \frac{26}{36}, & 8\le x<9,\\[4pt] \frac{30}{36}, & 9\le x<10,\\[4pt] \frac{33}{36}, & 10\le x<11,\\[4pt] \frac{35}{36}, & 11\le x<12,\\[4pt] 1, & x\ge 12. \end{cases} \]

8. Common Pitfalls & Tips

• PMF vs. PDF: Use a PMF only for discrete variables; a probability density function (PDF) belongs to continuous variables.
• Forgetting the square in variance: The formula \(\operatorname{Var}(X)=E(X^{2})-[E(X)]^{2}\) requires squaring the mean, not the expectation itself.
• Independence assumption: Binomial and geometric models assume independent trials. Verify independence before applying the formulas.
• Counting mistakes: Distinguish ordered (permutations) from unordered (combinations) outcomes when constructing a PMF.
• Exhaustive & mutually exclusive outcomes: Ensure that the list of possible values of \(X\) covers every outcome and that no two values can occur together.

9. Summary of Key Formulas

• PMF: \(p_X(x)=P(X=x)\).
• CDF: \(F_X(x)=P(X\le x)=\displaystyle\sum_{t\le x}p_X(t)\).
• Expectation: \(E(X)=\displaystyle\sum_{i}x_i\,p_X(x_i)\).
• Variance: \(\operatorname{Var}(X)=E(X^{2})-[E(X)]^{2}\).
• Linearity of expectation: \(E(aX+b)=aE(X)+b\).
• Scaling of variance: \(\operatorname{Var}(aX+b)=a^{2}\operatorname{Var}(X)\).
• Binomial: \(E(X)=np,\;\operatorname{Var}(X)=np(1-p)\).
• Geometric: \(E(X)=1/p,\;\operatorname{Var}(X)=(1-p)/p^{2}\).
• Poisson (preview): \(E(X)=\lambda,\;\operatorname{Var}(X)=\lambda\).
• Standardising a normal variable (P&S 5.5): \(Z=\dfrac{X-\mu}{\sigma}\sim N(0,1).\)