Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Probability and Statistics → Random Variables ↓


Discrete Random Variables


In the world of probability and statistics, random variables play a vital role in dealing with uncertainty and understanding data. A random variable is essentially a variable that takes on different values depending on the outcome of a random event. Random variables can be classified into two main categories: discrete random variables and continuous random variables. Here, we will focus specifically on discrete random variables.

What is a discrete random variable?

A discrete random variable is a type of random variable that can take on a countable number of possible values. This means that the values of a discrete random variable can be listed, even if the list is infinite. Examples of discrete values include whole numbers, such as 0, 1, 2, 3, and so on.

A key feature of discrete random variables is that there are distinct intervals between each possible value. This is in contrast to continuous random variables, which can take any value within a given range, with no intervals in between.

Examples of discrete random variables

To better understand discrete random variables, let's look at some examples:

Example 1: Throwing a six-sided dice

Consider the situation of rolling a fair six-sided dice. The possible outcomes are 1, 2, 3, 4, 5, and 6. In this case, the outcome of the dice roll is a discrete random variable because it can take only one of these six values. Each number represents one possible face of the dice that can fall face up.

Possible values of X (face value of the dice): 1, 2, 3, 4, 5, 6

The random variable X, in this scenario, is a discrete random variable that represents the value shown on the die.

Example 2: Number of heads when tossing a coin

Suppose you toss a fair coin three times, and you want to count the number of heads that come up. The possible outcomes for the number of heads are 0, 1, 2, or 3. Here, the random variable Y, which represents the number of heads, is discrete.

Possible values of Y (number of heads): 0, 1, 2, 3

In this example, Y is a discrete random variable representing the number of times heads occur in a sequence of coin tosses.

Example 3: Counting students in a class

Imagine a classroom where you are counting the number of students present. The possible values of the number of students are natural numbers, such as 0, 1, 2, 3, etc. This number is actually countable and represents a discrete random variable.

Possible values of z (number of students): 0, 1, 2, 3, ...

In this case, Z is a discrete random variable that counts the number of students in the room.

Probability mass function (PMF)

To better understand discrete random variables, we introduce the concept of the probability mass function, often abbreviated as PMF. The PMF is a function that gives the probability of each possible value of a discrete random variable.

The PMF is usually denoted as P(X = x), which is the probability that the random variable X is equal to a specific value x. For a PMF to hold, the following properties must hold:

  • The probability of each possible outcome is between 0 and 1, including:
    0 ≤ P(X = x) ≤ 1
  • The sum of all probabilities for all possible values must be equal to 1:
    ∑ P(X = x) = 1

Example: PMF for rolling a six-sided dice

For the die-rolling example, the PMF can be represented as follows:

p(x = 1) = 1/6
p(x = 2) = 1/6
p(x = 3) = 1/6
p(x = 4) = 1/6
p(x = 5) = 1/6
p(x = 6) = 1/6

The probability of each possible value (from 1 to 6) is 1/6, because the dice are fair, and each face has an equal chance of appearing.

Visualization of discrete random variables

To provide a visual representation of a discrete random variable, consider looking at the example of rolling a six-sided dice. A bar chart is a common way to visually represent the PMF of a discrete random variable.

1 2 3 4 5 6 Possibility

This chart shows the PMF for a six-sided dice, with each bar representing a probability of 1/6 for the values 1 to 6.

Cumulative distribution function (CDF)

Another important concept related to discrete random variables is the cumulative distribution function, abbreviated as CDF. The CDF gives the probability that a random variable X will be less than or equal to a particular value x, expressed as P(X ≤ x)

Example: CDF for rolling a six-sided dice

Let's consider the CDF for the die-rolling example:

p(x ≤ 1) = 1/6
p(x ≤ 2) = 2/6
p(x ≤ 3) = 3/6
p(x ≤ 4) = 4/6
p(x ≤ 5) = 5/6
P(x ≤ 6) = 6/6 = 1

This stepwise function accumulates probabilities as you move from one value to the next.

Expectation and variance of a discrete random variable

Expectation (mean)

The expectation or mean of a discrete random variable is a measure of its central tendency. It is a weighted average of the possible values that the random variable can take, with the weights being their probabilities. The expectation is denoted as E(X) or μ.

The formula for calculating the expectation of a discrete random variable X is:

e(x) = ∑ [x * p(x = x)]

Example: Expectation of rolling a six-sided dice

For the die-rolling example, the expectation is calculated as follows:

e(x) = 1 * (1/6) + 2 * (1/6) + 3 * (1/6) + 4 * (1/6) + 5 * (1/6) + 6 * (1/6) = 21/6 = 3.5

If the dice are thrown many times the expected value is 3.5, the average result.

Variance

The variance of a discrete random variable measures the dispersion or spread of its possible values. It is the expected value of the squared deviation from the mean. The variance is represented as Var(X) or σ2.

The formula for variance is:

Var(X) = E[(X - μ)2 ] = ∑ [(x - μ)2 * P(X = x)]

Example: Variance for rolling a six-sided dice

For the die-rolling example where μ = 3.5, the variance is calculated as follows:

Var(x) = [(1 - 3.5)2 * (1/6)] + [(2 - 3.5)2 * (1/6)] +
         [(3 – 3.5)2 * (1/6)] + [(4 – 3.5)2 * (1/6)] +
         [(5 – 3.5)2 * (1/6)] + [(6 – 3.5)2 * (1/6)]
       = [(-2.5)2 * (1/6)] + [(-1.5)2 * (1/6)] +
         [(-0.5)2 * (1/6)] + [(0.5)2 * (1/6)] +
         [(1.5)2 * (1/6)] + [(2.5)2 * (1/6)]
       = 2.92

The variance of 2.92 indicates the average square deviation of the dice rolls from the expectation.

Conclusion

Discrete random variables are fundamental to probability and statistics and are used to model many real-world situations where outcomes are countable. Understanding how to work with discrete random variables, including calculating their probability mass function, cumulative distribution function, expectation, and variance, provides a solid foundation for analyzing random events and measuring uncertainty.

By understanding these concepts, they can be applied to a variety of areas, ranging from designing games of chance to decision-making processes involving probability evaluations.


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Discrete Random Variables

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