E(X) & Var(X) (Continuous)

What are E(X) and Var(X)?

E(X)is the expected value, or mean, of a random variable X
- E(X) is the same as the population mean so can also be denoted by µ
Var (X) is the variance of the continuous random variable X
- Standard deviation is the square root of the variance

How do I find the mean and variance of a continuous random variable?

The mean, for a continuous random variable X is given by

$E (X) = \int_{- \infty}^{\infty} x f (x) d x$

This is equivalent to $Σ x P (X = x)$ for discrete random variables
If the graph of $y = f (x)$ has axis of symmetry, x = a , then E(X) = a

The variance is given by

$Var (X) = \int_{- \infty}^{\infty} x^{2} f (x) d x - {[E (X)]}^{2}$

This is equivalent to $Σ x^{2} P (X = x) - {[E (X)]}^{2}$ for discrete random variables
Be careful about confusing $E (X^{2})$ and ${[E (X)]}^{2}$
- $E (X^{2}) = \int_{- \infty}^{\infty} x^{2} f (x) d x$ “mean of the squares”
- ${[E (X)]}^{2} = {[\int_{- \infty}^{\infty} x f (x) d x]}^{2}$ “square of the mean”
- If you are happy with the difference between these and how to calculate them the variance formula becomes very straightforward

$Var (X) = E (X^{2}) - {[E (X)]}^{2}$

How do I calculate E(g(X))?

$E (g (X)) = \int_{- \infty}^{\infty} g (x) f (x) d x$
In particular:
- $E (X^{2}) = \int_{- \infty}^{\infty} x^{2} f (x) d x$ as seen above

Worked example

A continuous random variable, $X$ , is modelled by the probability distribution function $f (x)$ , such that

$f (x) = \{\begin{cases} 1.5 x^{2} (1 - 0.5 x) & 0 \leq x \leq 2 \\ 0 & otherwise \end{cases}$

(i)

Find

E (X)

(ii)

Find

Var (X)

(i)

Find

E (X)

2-3-2-cie-fig1-we-solution_a

(ii)

Find

Var (X)

2-3-2-cie-fig1-we-solution_b

Examiner Tip

A sketch of the graph of y = f(x) can highlight any symmetrical properties which can help reduce the work involved in finding the mean and variance
Take care with awkward values and negatives – use the memory features on your calculator and avoid rounding until your final answer (if rounding at all!)

E(X) & Var(X) (Continuous) (CIE A Level Maths: Probability & Statistics 2): Revision Note

E(X) & Var(X) (Continuous)

What are E(X) and Var(X)?

How do I find the mean and variance of a continuous random variable?

How do I calculate E(g(X))?

Worked example

Examiner Tip

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1. Statistical Sampling

1.1 Sampling & Data Collection

1.1.1 Sampling & Data Collection

1.2 Sampling & Estimation

1.2.1 Unbiased Estimates

1.2.2 Distribution of Sample Means

1.2.3 Confidence Intervals

2. Statistical Distributions

2.1 Poisson Distribution

2.1.1 The Poisson Distribution

2.2 Linear Combinations of Random Variables

2.2.1 Linear Combinations of Random Variables

2.3 Continuous Random Variables

2.3.1 Probability Density Function

2.3.2 E(X) & Var(X) (Continuous)

2.3.3 Continuous Uniform Distribution

2.4 Working with Distributions

2.4.1 Choosing Distributions

2.4.2 Approximations of Distributions

3. Hypothesis Testing

3.1 Hypothesis Testing

3.1.1 Hypothesis Testing

3.1.2 Type I & Type II Errors

3.2 Hypothesis Testing (Discrete Distribution)

3.2.1 Binomial Hypothesis Testing

3.2.2 Poisson Hypothesis Testing

3.3 Hypothesis Testing (Normal Distribution)

3.3.1 Normal Hypothesis Testing

Author: Paul