Mean measures central value; variance measures spread around it.
Expectation E[X]=Σx p(x) or ∫x f(x)dx is the long-run average; variance Var=E[X²]-E[X]² measures spread, and the standard deviation is its square root. E is linear; variances of independent variables add.
A fair die has mean 3.5 and variance 35/12≈2.92.
Expectation and variance summarise a random quantity in two numbers, which is the first step in reliability and tolerance analysis.
Mean measures central value; variance measures spread around it.
Expectation and variance summarise a random quantity in two numbers, which is the first step in reliability and tolerance analysis.
Variance measures how spread out values are around the mean, so two processes with the same average can have very different reliability.
Their variances add, which is why independent error sources accumulate in a sum.
Typical applications include estimating the average annual energy from a wind turbine with random speed, computing the spread in strength of a batch of steel rods and predicting the average cost of repairs with random failure times.
Study "Random variables" first (topic 79 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
E[aX+b]=aE[X]+b. Var(aX)=a²Var(X). Covariance measures joint variation.
A fair die has mean 3.5 and variance 35/12≈2.92.
The next topic is "Binomial and Poisson distributions", which builds directly on this one.
It is classed as Advanced in Engineering Mathematics (topic 80 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.
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