CHAPTER 4 EXPECTATION. CHAPTER 4. Overview. ● The Expectation of a R. V. ● Properties of Expectation ● Variance ● Moments ● The Mean and the Median ● Covariance and Correlation ● Conditional Expectation ● The Sample Mean. Section 4.1 The Expectation of a Random Variable.
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● The Expectation of a R. V.
● Properties of Expectation
● The Mean and the Median
● Covariance and Correlation
● Conditional Expectation
● The Sample Mean
The Expectation of a Random Variable
= ∑i (1 ∕ N ) * xi
E[Y] = (-3)2(0.4)+(-1)2(0.1)+(1)2(0.2)+(3)2(0.3)
= 36/10 + 1/10 + 2/10 + 27/10 = 66/10 = 6.6
E[Xn] = -∞∫∞xnf(x)dx
density in order to calculate its expected value.
Properties of Expectations
(the results are gathered without proof in the next slide)
See Class Notes for definitions, properties and examples of the variance and standard deviation.
Section 4.4 the variance and standard deviation.
We recall the definition of moments of distributions.
Also we explore a new technique that makes that makes it simple to find the mean and variance of certain distributions whose moments are otherwise hard to calculate.
This technique is called the method of the moment generating function.
Ψ(t) = E[etX]
That is, in particular Ψ’(0)=E[X]; Ψ’’ (0)=E[X2]; etc.
is ΨX(t) = i=1∏nΨi(t).