By Jiruse M., Machek J.
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Extra resources for A Bayesian estimate of the risk of Tick-Borne deseases
Expectation and Variance 49 Proof. (a) Let E(X) = μ. We then write var(aX + b) = = = = E((aX + b)2 ) − (E(aX + b))2 E(a2 X 2 + 2abX + b2 ) − (aμ + b)2 a2 E(X 2 ) + 2abμ + b2 − a2 μ2 − 2abμ − b2 a2 E(X 2 ) − a2 μ2 = a2 var(X). (b) var(X + Y ) = E((X + Y − E(X + Y ))2 ) = E((X − E(X))2 + (Y − E(Y ))2 + 2(XY − E(X)E(Y ))) = var(X) + var(Y ) + 2(E(XY ) − E(X)E(Y )). 21. The quantity E(XY )−E(X)E(Y ) which appears in (b) is called the covariance of X and Y , and denoted by cov(X, Y ). 22. Usually the covariance of X and Y is deﬁned as cov(X, Y ) = E((X − E(X))(Y − E(Y ))).
In that example, even when we know that a family has at least one boy, when we then actually see a boy opening the door, this new information does change the conditional probability that the family has two boys. The bare fact that a boy opened the door, makes it more likely that there are two boys. Similarly, the fact that the ﬁrst person to be screened has the DNA proﬁle, makes it more likely that there are more such persons. 17. Method (1) above can be made correct by taking into account the so called size bias which we tried to explain above.
Xd and similarly for the other marginals. In words, we ﬁnd the mass function of X1 by summing over all the other variables. Proof. 11, where we take A to be the event that X1 = x1 and the Bi ’s all possible outcomes of the remaining coordinates. 54 Chapter 2. 6. Provide the details of the last proof. 7. 3.
A Bayesian estimate of the risk of Tick-Borne deseases by Jiruse M., Machek J.