Continuous Probability Distributions (HSC SSCE Mathematics Advanced): Quizzes

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Mean and Variance of a Distribution
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What replaces summation when moving from discrete to continuous probability distributions?

Integration

What is the formula for mean E(X)E(X) of a continuous distribution with pdf f(x)f(x) on [a,b][a,b]?

abxf(x)dx\int_a^b xf(x) \, dx

What does the mean of a continuous distribution represent?

The balancing point

What does variance measure in a probability distribution?

How spread out the distribution is

Which variance formula typically involves simpler algebra?

E(X2)μ2E(X^2) - \mu^2

What is the formula for standard deviation σ\sigma in terms of variance?

Var(X)\sqrt{\text{Var}(X)}

If XX measures distance in metres, what units does standard deviation have?

Metres

For a uniform distribution on [0,8][0, 8], where is the mean located?

Exactly halfway at 4

For f(x)=118xf(x) = \frac{1}{18}x on [0,6][0,6], what is the mean?

4

Do both variance formulas give the same answer?

Yes, exactly the same

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