Unformatted text preview: er x as
follows 1.
2.
3. as
as
is a nondecreasing function of 5 6 Properties of the pdf Probability Density Function (pdf)
• Continuous random variable
• Nonnegative function f defined
on the real line.
• For every real interval 1.
2.
fX(x) 3.
• First two properties follow from the
axioms of probability.
• Integrate: x
x
x + dx P x X x dx f X ( x)dx
7 8 Example: Spin the Pointer Uniform Distribution 1 , a xb f X ( x ) b a
0,
elsewhere 1 1/(b a) a x1 x2 b 2 2 1 x • Probability of value in any subinterval of
is proportional to its length.
• Area of rectangle must be unity. 1
2
2 1
MATLAB >> rand(m, n) % a=0, b=1, m by n
9 10 Expectation of a Random Variable Density & Distribution • Expected value or mean of .
• Justified by relative frequency
• Discrete . • Continuous 11 12 Properties of Expectation Function of a Random Variable
• Expected value of the function
• Discrete: • Continuous: 13 14 Moments Variance moment: expectation of
power
• First moment is the mean
• Second moment is the mean square
• Discrete • Second moment about the mean • Continuous • Continuous Standard Deviation: square root of variance
• Discrete 15 16 Variance Property: Proof
Properties of the Variance = − = + = + = • Uncorrelated: − − − = ( ), − 17 Example: Uniform Distribution 18 Characteristic Function Find the mean and variance =
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 Fall '13
 M.SamiFadali
 Electrical Engineering, Normal Distribution, Probability theory, CDF

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