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Px vs. 2 are iid Exponential(θ) rv’s (by A164) The Exponential(θ) rv is the special case of the Gamma(p,θ) distribution with density with p = 1 f (x θ, p) = θ p x e p−1 −θx, 0 < x < ∞ Γ(p) Theorem B23 If X 1 and X 2 are independent random variables with Γ(p,λ) and Γ(q,λ) distributions, Y 1 = X 1 X 2 and Y. Q w G º ± e U q r q s l h } ² \ ` h O t @ Í p. L @ ® @ b P X A e P X A y P X i ¤ Ê j @ Ö Ô F ¤ í Ú F Û è Ô F ¤ Û è ¼ i a ¶ j ¤ ã \ Ò ¤ Û è ¼ i p ¶ j ð t è z i ¤ ú Ô S Ì j F i ¼ Ú o ï j P R P O P î Õ ¤ (C) 11 `14.

2 So we can can write p(x) as a linear combination of p 0;p 1;p 2 and p 3Thus p 0;p 1;p 2 and p 3 span P 3(F)Thus, they form a basis for P 3(F)Therefore, there exists a basis of P 3(F) with no polynomial of degree 2 Exercise 2 Prove or give a counterexample If v. Title GodkendtSpildevand21xlsx Author sb Created Date 11/17/ AM. O s v } v s D d } ñ & } µ ' } µ W P ò W } Ç W P ó r í ì µ } u ^ À W } Ç W P í í r í ò.

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P(k) = P(X = k) given by p(1) = p, p(0) = 1−p, p(k) = 0, otherwise Thus X only takes on the values 1 (success) or 0 (failure) A simple computation yields E(X) = p Var(X) = p(1−p) M(s) = pes 1−p Bernoulli rvs arise naturally as the indicator function, X = I{A}, of an event A, where I{A} def= ˆ 1, if the event A occurs;. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers Visit Stack Exchange. 3 (8 points) Let T 2L(V);V a vector space over R;p(x) a polynomial in R, and a2R Show that the following statement is false ais an eigenvalue of p(T) if and only if a= p( ) for some eigenvalue of T Rather than simply providing a counterexample, prove as strong a statement as possible Solution.

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