1. Random Variables (1)
A random variable (also known as a stochastic variable), x, is a quantity such as strength, size, or weight, that depends upon a the outcome of a random experiment. The domain of all possible outcomes of the experiment is called the sample space. Consider an experiment involving the throw of two dice. The sample space is:
1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
4,1
4,2
4,3
4,4
4,5
4,6
5,1
5,2
5,3
5,4
5,5
5,6
6,1
6,2
6,3
6,4
6,5
6,6
Sample Space
Now consider the probability of getting a specific number on a given throw: x 2 3 4 5 6 7 8 9 10 11 12
f(x)
1/36
2/36
3/36
4/36
5/36
6/36
Probability Distribution

2. Random Variables (2)
Frequency distribution
A plot of of the probability distribution data is called the frequency distribution. The Probability function itself, p(x), is also called the frequency distribution or the probability density.
We can also define a cumulative probability distribution, and a corresponding cumulative probability function, F(x):
A random variable can be classified as discrete, or continuous.
Cumulative frequency distribution x 2 3 4 5 6 7 8 9 10 11 12
f(x)
1/36
3/36
6/36
10/36
15/36
21/36
26/36
30/36
33/36
35/36
36/36
Cumulative Probability Distribution

3. Statistical Parameters (1)
The total number of elements associated with a random variable (the population) may be large or infinite, so a small group (sample) is is used. To quantify a distribution we need a measure of central value. An arithmetic mean can be defined for both a sample and a population. For N elements:
Sample mean value
Population mean value
The mode (value that occurs most frequently) and median (middle value for an odd number of cases; mean of the two middle values if there is an even number of cases) can also be used as measures of central value.
We also need a measure of the dispersion of the distribution. The deviation from the mean is given by:

4. Statistical Parameters (2)
The sum of the deviations is zero, so we define a sample variance based upon the square of the deviations:
Sample standard deviation
A population standard deviation is denoted by s.
The ratio of the standard deviation to the mean is called the coefficient of variation (COV):
Distribution of strength properties of hot-rolled UNS G10350 steel. (a) Tensile strength, (b) yield strength.

5. Gaussian Distribution
The Gaussian, or normal distribution provides an excellent representation of many population distributions associated with engineering phenomena. The distribution is a function of its mean value and standard deviation:
standardized variable
Shape of the Gaussian distribution for (a) a small standard deviation and (b) a large standard deviation
cumulative probability function for a Gaussian distribution
x is a normally distributed variable with a mean of mx and a standard deviation of sx. This is equivalent to the mean mx multiplying a normal variable with a mean of 1.0 and a standard deviation of Cx=sx/mx.
tabulated in Table E-10