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Continuous Random Variables

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  Edexcel IAL Statistics 2 (WST02)

Continuous Random Variables

The big idea: For a continuous variable, probability is area under the probability density function; the cumulative distribution function is that area accumulated from the left.

Summary — What This Chapter Covers
  • f(x) ≥ 0 and ∫f(x) dx = 1 over the whole range.
  • F(x) = ∫ from the lower limit to x of f(t) dt; f(x) = F′(x).
  • E(X) = ∫xf(x) dx, Var(X) = ∫x2f(x) dx − [E(X)]2.
  • Median: F(m) = 0.5. Mode: where f(x) is greatest.
1. Probability Density Functions
pdf

Area is probability

Use ∫f(x) dx = 1 to find unknown constants.

Worked Example

f(x) = kx for 0 ≤ x ≤ 2 (0 otherwise). Find k.

2. Cumulative Distribution Functions
cdf

Accumulated area

Define F(x) on every interval, including F(x) = 0 below the range and 1 above it.

Worked Example

For f(x) = ½x on 0 ≤ x ≤ 2, find F(x) and the median.

3. Mean and Variance
Expectation

E(X) and Var(X)

Integrate x f(x) and x2 f(x) over the range.

Worked Example

Find E(X) and Var(X) for f(x) = ½x on 0 ≤ x ≤ 2.

4. Mode, Median and Quartiles
Averages

Reading the shape

Memorise
Total area

∫f dx = 1

cdf

f = F′

Variance

E(X²) − E(X)²

Median

F(m) = 0.5

Common Mistakes
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More Statistics 2 topics

The Binomial and Poisson Distributions
The binomial distribution
Probability density functions
The continuous uniform distribution
Populations, samples and sampling distributions
The Poisson distribution
Cumulative distribution functions
Normal approximations to the binomial and Poisson
Critical regions and significance levels
Approximating a binomial by a Poisson