3.2 Review of Probability and Statistics

3.2.4 Expectations

Measurable Outcome 3.5, Measurable Outcome 3.6, Measurable Outcome 3.10

Think back to the fact that the thickness of a turbine blade in an engine is a random variable. Even though we recognize the randomness of the blade thickness, this does not imply that the nominal value of the blade thickness that we would have used in a deterministic analysis is without any meaning. We understand that the nominal value of the blade thickness is the thickness we expect the manufacturer to produce. This nominal value provides us with a centre to base our further calculations relating to the random variable.

The nominal value is an example of an expected value of a random variable. Expected values of random variables are deterministic variables that help us understand the random variable.

Expected Value and Mean

Given a random variable \(X\) with PDF \(f_ X(x)\), the expected value is defined as,

\[E\{ X\} \equiv \int _{-\infty }^{+\infty } x f_ X(x)\, dx\]

(3.13)

The expected value of \(X\) is also known as the mean value. We will also use the symbol \(\mu _ X\) for the expected value of \(X\).

Variance and Standard Deviation

The variance is a measure of the variability in \(X\) about its mean value. The variance of \(X\) is defined as,

\[\sigma _ X^2 \equiv \int _{-\infty }^{+\infty } (x - \mu _ X)^2 f_ X(x)\, dx.\]

(3.14)

The value \(\sigma _ X\) is known as the standard deviation of \(X\). A frequently used relationship exists between the mean and variance,

\[\sigma _ X^2 = E\{ X^2\} - \mu _ X^2.\]

(3.15)

Try proving this!

Percentiles

The \(u\) percentile of \(X\) is the smallest number \(x_ u\) such that,

\[u = P\{ X \leq x_ u\} = F(x_ u).\]

(3.16)

Note, since \(u\) is a probability, its range is \(0 \leq u \leq 1\).