A random variable assigns numerical values to the outcomes of a random phenomenon. In daily situations, outcomes can either be counted, such as the number of heads obtained when tossing a coin, or measured over an interval, such as the waiting time for a bus. This note covers the classification of random variables into discrete and continuous types, and explores how the probability distribution of a discrete random variable can be determined, tabulated, and graphed.
Tree diagram, probability distribution table, and stick line graph of \(P(X = r)\) against \(r\).
Countable values such as \(X = \{0, 1, 2, 3\}\). Each value has a distinct probability mass, shown by separate vertical lines on a stick graph.
Measured values within an interval such as \(X = \{x : a \le x \le b\}\). The probability is represented by a shaded region along a number line.
A random variable converts outcomes of chance events into numbers.
A random variable is a variable with numeric values that can be determined from a random phenomenon. It is represented by a capital letter, typically \(X\), while its specific values are represented by lowercase letters such as \(r\) or \(x\). Possible values can be expressed in set notation, for example, \(X = \{0, 1, 2\}\).
In an experiment with a sample space \(S\), each outcome is mapped to a real number. For example, in two basketball matches where outcomes are win (\(W\)), lose (\(L\)), or draw (\(D\)), the sample space is \(S = \{WW, WL, WD, DW, DL, DD, LW, LD, LL\}\). If \(X\) is the number of wins, \(X\) maps outcomes to values in \(\{0, 1, 2\}\).
A discrete random variable has a countable number of values, usually zero and positive integers. It is written in set notation by listing elements: \(X = \{r : r = 0, 1, 2, 3, \dots, n\}\).
A continuous random variable takes values that are not solely integers, but lie continuously within a certain interval. It is obtained by measurement and written using an inequality: \(Y = \{y : a \le y \le b\}\).
A probability distribution lists all possible values \(r\) of a discrete random variable \(X\) together with their corresponding probabilities \(P(X = r)\).
Each probability is non-negative and between \(0\) and \(1\): \(0 \le P(X = r_i) \le 1\). The sum of all probabilities is equal to \(1\): \(\sum_{i=1}^{n} P(X = r_i) = 1\).
Probability of \(r\) successes in \(n\) independent trials
where \(p\) is the probability of success and \(q = 1 - p\) is the probability of failure.
Identify the random variable \(X\) and its possible values.
Draw a tree diagram to list all possible outcomes.
Multiply probabilities along each branch and add probabilities for the same value of \(X = r\).
Verify that \(\sum P(X = r) = 1\), then construct the table and graph.
Answer in your mind, then press "Check Answer".
A fair coin is tossed three times. Let \(X\) represent the number of times getting 'heads'. State all possible values of \(X\) in set notation and determine whether \(X\) is discrete or continuous.
The random variable \(X = \{0, 1, 2, 3, 4\}\) has the probability distribution shown below. If \(p = 2q\), find the values of \(p\) and \(q\).
A box contains 2 basketballs and 4 footballs. A ball is drawn at random, its type is recorded, and it is returned to the box. This process is repeated 2 times. If \(X\) represents the number of basketballs drawn, find \(P(X = 0)\), \(P(X = 1)\), and \(P(X = 2)\), and show that \(\sum P(X = r) = 1\).
All variables measured in numbers are discrete.
If a variable is measured along a continuous scale and can take any value within an interval \(a \le x \le b\), it is a continuous random variable, not discrete.
The probability of an outcome can be greater than 1 or negative.
Every probability \(P(X = r)\) must strictly satisfy \(0 \le P(X = r) \le 1\).
Writing the set notation of a continuous random variable by listing isolated numbers.
Continuous random variables cannot be listed as individual elements because there are infinitely many values between any two points; they must be stated as an interval such as \(\{x : a \le x \le b\}\).
Objective: Identify discrete and continuous random variables, verify probability distribution properties, and match key terms.
1 Which of the following represents a continuous random variable?
2 The discrete random variable \(X\) has the probability function given by \(P(X = 0) = 0.2\), \(P(X = 1) = 0.35\), \(P(X = 2) = 0.3\), and \(P(X = 3) = k\). What is the value of \(k\)?
3 For any discrete probability distribution, the sum of all individual probabilities \(\sum_{i=1}^{n} P(X = r_i)\) must equal ________.
4 A random variable that takes values lying within a continuous interval rather than countable values is known as a ________ random variable.
Drag each situation or definition to its correct description or classification. On a phone, tap a term first, then tap the matching box.
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