Download Lecture notes, week 2

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
5/36
4/36
3/36
2/36
1/36
Probability Distribution
ME 422 – Machine Design I
Statistical Considerations, Slide 1
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):
Cumulative frequency distribution
x
F( x )   f ( x )dx


F()   f ( x )dx  1.0

A random variable can be classified as
discrete, or continuous.
x
2
3
4
f(x)
1/36
3/36
6/36
5
6
7
8
9
10
11
12
10/36 15/36 21/36 26/36 30/36 33/36 35/36 36/36
Cumulative Probability Distribution
ME 422 – Machine Design I
Statistical Considerations, Slide 2
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:
x
x1  x 2  x3    xN 1 N
 xj
N
N j1
x1  x 2  x 3    xN 1 N
x 
 xj
N
N j1
– 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:
xi  x
ME 422 – Machine Design I
Statistical Considerations, Slide 3
Statistical Parameters (2)
The sum of the deviations is zero, so
we define a sample variance based
upon the square of the deviations:
Sut  86.0 kpsi
sSut  4.94 kpsi
( x1  x )2  ( x1  x )2  ( x1  x )2    ( x1  x )2
N 1
1 N

( x j  x )2

N  1 j1
s2x 
 1 N

sx  
( x j  x )2 

 N  1 j1

1/ 2
– Sample standard
deviation
Sy  49.5 kpsi
A population standard deviation is
denoted by s.
sSy  5.36 kpsi
The ratio of the standard deviation to
the mean is called the coefficient of
variation (COV):
Cx 
sx
x
ME 422 – Machine Design I
Distribution of strength properties of hot-rolled UNS
G10350 steel. (a) Tensile strength, (b) yield strength.
Statistical Considerations, Slide 4
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:
 1  x   2 
1
x  
f (x) 
exp  
2
 2  s x  
 N( x , s x )
z
x  x
sx
F( z  )  
z

– standardized variable
 u2 
1
exp   du  ( z  )
2
 2
– cumulative probability
function for a Gaussian
distribution
( z )
tabulated in Table E-10
ME 422 – Machine Design I
Shape of the Gaussian distribution for (a) a small
standard deviation and (b) a large standard deviation
x  N(x , sx )  xN(1,Cx )
– x is a normally distributed variable with a mean of x 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/x.
Statistical Considerations, Slide 5