1ME 233 Advanced Control II

Lecture 3
Introduction to Probability Theory(ME233 Class Notes pp. PR1-PR3)

2Outline• Continuous random variable• CDF, PDF, expectation and variance• Uniform and normal PDFs

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Continuous random variable
A continuous-valued random X variable takes
on a range of real values• For the probability space,• A random variable X is a mapping

Example:• An experiment whose outcome is a real
number, e.g. measurement of a noisy
voltage.

X  [Vmin ,Vmax ]4Cumulative Distribution Function
Cumulative distribution function (CDF) associatedwith the random variable X :The probability that the random variable X
will be less than or equal to the value x

5Properties of the cumulative distribution

10a

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Properties of the cumulative distribution

1.

2.

3.

is a monotone non decreasing

4.

is left-continuous

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Probability Density Function
For a differentiable cumulative distribution function,

Define the probability density function (PDF),

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Probability Density Function

Interpretation:

for small
Loosely interpret this as the probability that X takes
a value close to x

9Probability Density Function

By the Fundamental Theorem of Calculus

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Probability Density Function

ab

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Probability Density Function
Property:

because12Expectation
The expected value of random variable X is:This is the average value of X.
It is also called the mean of X
or the first moment of X

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Expected value - notation

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Expected value of a function
f : real valued function of random variable X

The expected value of Y is

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Variance
The variance of random variable X is:

whereIs called the standard deviation of X

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Variance

where17Proof18

Uniform Distribution
A random variable X which is uniformly distributed
between xmin and xmax has the PDF:

p (x)
X

matlab function: rand

1xmaxxminxminxmaxx

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Summing independent uniformly distributed randomvariables• Let X and Y be 2 independent uniformly
distributed variables between [0,1]• The random variable

Z  X Y
• is not uniformly distributed

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Summing independent uniformly distributed randomvariables• Let X and Y be 2 independent uniformly
distributed variables between [0,1]Z  X Y
105 randomsamples of Z
Histogram ofZ withnormalizedarea

X=rand(1,1e5);
Y=rand(1,1e5);
Z=X+Y;[freqZ,cent]=hist(Z,100);
bin_width=(cent(100)-cent(1))/99;
area = sum(freqZ)*bin_width;bar(centers,freqZ/area)
xlabel('z')
ylabel('F_Z(z)')

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Summing independent uniformly distributed randomvariables• Let X and Y be 2 independent uniformly
distributed variables between [0,1]Z  X Yz22

Summing a very large number of random variables• Let X1,…,X1000 be independent uniformly
distributed variables between [0,1]z23Gaussian (Normal) Distribution

Normal distribution

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History of the Normal Distribution
From Wikipedia:• The normal distribution was first introduced byde Moivre in an article in 1733 in the contextof approximating certain binomial distributions
for large n.
• His result was extended by Laplace in his
book Analytical Theory of Probabilities (1812),
and is now called the theorem of de MoivreLaplace.
• Laplace used the normal distribution in theanalysis of errors of experiments.

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History of the Normal Distribution
From Wikipedia:• The important method of least squares wasintroduced by Legendre in 1805.
• Gauss, who claimed to have used the method since
1794, justified it rigorously in 1809 by assuming anormal distribution of the errors.
• That the distribution is called the normal or Gaussian
distribution is an instance of Stigler's law of eponymy:"No scientific discovery is named after its originaldiscoverer."Supplemental Material(You are not responsible for this…)• Laplace transform of normal PDF• Proof of the central limit theorem26

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Laplace transform of normal PDF

where, after “completing the squares”,

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Laplace transform of normal PDF
substituting,= 1 (area under a PDF = 1)

Fourier transform:

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Proof of the central limit theorem
Let X1, X2,... be independent random variables each withmean mx and variance σx2 and define the sequence

wherenotice that

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Proof of the central limit theorem
The moment generating function of Zn isby the Taylor series expansion of ex

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Proof of the central limit theorem
notice that, as n →∞ the approximation is exact

Moreover, the PDF and moment generating function of anormally distributed random variableareand32

Proof of the central limit theorem
Therefore, sinceThen, taking the inverse Fourier transform we obtainand