Practice Random variables: expectation and standard deviation
A random variable assigns a numerical value to each outcome of an experiment.
Its distribution and expected value describe the possible values and their long-run average.
Exercises with solutions
A random variable assigns a number to each outcome. The expected value E(X) is the long-run mean: multiply each value by its probability and add the products. The variance measures the mean squared deviation, and the standard deviation σ is its square root.
Exercise 1
A die shows its score. What is the expected value?
Show solution and working
3.5
Every score has probability 1/6. (1+2+3+4+5+6) ÷ 6 = 21 ÷ 6 = 3.5.
Exercise 2
What does the expected value express?
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the long-run mean
Over very many repetitions the average approaches the expected value. It need not be a possible value itself, since 3.5 cannot be rolled.
Exercise 3
A random variable has E(X) = 10 and σ = 2. In which range do almost all values lie?
Show solution and working
roughly between 4 and 16
By the sigma rule about 99.7 % of values lie within E ± 3σ, so between 10 − 6 = 4 and 10 + 6 = 16.
Exercise 4
How do you compute the expected value from a distribution table?
Show solution and working
multiply each value by its probability, then add
E(X) = x₁ · P(x₁) + x₂ · P(x₂) + … until all values are included.
Common mistake
Simply adding the probabilities instead of weighting them. Each value must be multiplied by its own probability, otherwise you do not get the expected value.
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