STAT1008 Chap.7 Discrete Random Variables and Expected Value
Discrete Random Variables and Expected Value
Discrete Random Variables and Expected Value defines a random variable as a number attached to the outcome of a random experiment and a discrete distribution as a list of values with probabilities summing to 1. It computes the expected value as a probability-weighted average, the variance through E(X squared) minus the squared mean, and the standard deviation, then applies the rules for linear transformations and sums of independent variables.
It separates the expected value, a parameter, from the sample mean, a statistic. Typical questions give a distribution table, or a rule that defines the probabilities, and ask for its mean and standard deviation, a probability for an inequality, and the effect of a linear change such as a price or a fixed fee. An interpretation in context closes each part.
What this chapter covers
- 01
Random variables, discrete and continuous
- 02
Requirements for a valid distribution
- 03
Expected value as a long-run average
- 04
Variance and standard deviation of a random variable
- 05
Linear transformations aX + b
- 06
Sums of independent random variables
Worked example · free
Expected number of textbooks bought
- 1Check: 0.10 + 0.25 + 0.35 + 0.20 + 0.10 = 1, so the distribution is valid.
- 1E(X) = 0 + 0.25 + 0.70 + 0.60 + 0.40 = 1.95 textbooks.
- 2E(X²) = 5.05, so Var(X) = 5.05 − 1.95² = 1.2475 and SD(X) = 1.1169.
- 1P(X ≥ 3) = P(3) + P(4) = 0.20 + 0.10 = 0.30.
Key terms
- Random variable
- A random variable assigns a numerical value to every outcome of a random experiment.
- Expected value
- The expected value is the probability-weighted average of a random variable’s possible values.
- Probability distribution
- A probability distribution lists every possible value of a discrete random variable with its probability.
- Discrete random variable
- A discrete random variable takes values in a countable set, such as whole-number counts.
- Variance of a random variable
- The variance of a random variable is the probability-weighted average squared distance of its values from the expected value.
Discrete Random Variables and Expected Value FAQ
Is the expected value the same as the sample mean?
No. The expected value is computed from the probability distribution before any data are seen and is a parameter. The sample mean is computed from observed data, weights each observation equally and serves as an estimate of the expected value.
Does adding a fixed cost change the standard deviation?
No. Adding a constant shifts every value by the same amount, so the mean moves but the spread stays the same. Multiplying by a constant changes both: the mean and the standard deviation are multiplied by it.
Can I add standard deviations of two independent variables?
No. For independent variables the variances add, and the standard deviation of the sum is the square root of that total. Adding standard deviations directly overstates the spread of the sum.
Can the expected value be a value the variable never takes?
Yes. The expected value is a long-run average weighted by probability, so it can fall between possible values. An expected 1.95 textbooks per student is perfectly valid even though no student buys a fraction of a book.
How do I find a missing probability in a distribution table?
Use the rule that the probabilities of a valid discrete distribution sum to 1. Subtract the known probabilities from 1. If the probabilities are given in terms of an unknown constant, set their total equal to 1 and solve for it.
Exam move
Build a column layout for every distribution: x, P(x), xP(x) and x²P(x), then sum the columns. Do this by hand for three or four distributions until it takes under two minutes. Then practise the transformation rules on revenue and cost questions, writing the independence assumption whenever you add variances, and finish each answer with an interpretation in units.
Before the final, rework the textbook example on this page from a blank sheet and compare your columns with the printed ones. Pair every distribution you practise with one transformation question, for example revenue from units sold or the total for two independent buyers.
Writing the rule you used beside each step, with the independence assumption stated where variances are added, is the clearest evidence of method a marker can see.
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