Statistical Considerations, Slide 1ME 422 – Machine Design I

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

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 5/36 4/36 3/36 2/36 1/36

Sample Space

Probability Distribution

Statistical Considerations, Slide 2ME 422 – Machine Design I

Random Variables (2)

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

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.

Frequency distribution

Cumulative frequency distribution

0.1dx)x(f)(F

dx)x(f)x(Fx

Statistical Considerations, Slide 3ME 422 – Machine Design I

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:

N

1jj

N321 xN

1

N

xxxxx

N

1jj

N321x x

N

1

N

xxxx

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:

xxi

– Sample mean value

– Population mean value

Statistical Considerations, Slide 4ME 422 – Machine Design I

Statistical Parameters (2)

The sum of the deviations is zero, so we define a sample variance based upon the square of the deviations:

N

1j

2j

21

21

21

212

x

)xx(1N

1

1N

)xx()xx()xx()xx(s

Distribution of strength properties of hot-rolled UNS G10350 steel. (a) Tensile strength, (b) yield strength.

2/1N

1j

2jx )xx(

1N

1s

kpsi94.4s

kpsi0.86S

utS

ut

kpsi36.5s

kpsi5.49S

Sy

y

A population standard deviation is denoted by .

The ratio of the standard deviation to the mean is called the coefficient of variation (COV):

x

xxC

– Sample standarddeviation

Statistical Considerations, Slide 5ME 422 – Machine Design I

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:

),(N

x

2

1exp

2

1)x(f

xx

2

x

x

x

xxz

)z(du2

uexp

2

1)z(F

z 2

)z( tabulated in Table E-10

– standardized variable

– cumulative probabilityfunction for a Gaussian distribution

Shape of the Gaussian distribution for (a) a small standard deviation and (b) a large standard deviation

)C,1(N),(Nx xxxx

– x is a normally distributed variable with a mean of x and a standard deviation of x. This is equivalent to the mean mx multiplying a normal variable with a mean of 1.0 and a standard deviation of Cx=x/x.