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11 COMPUTING PROBABILITIES AND EXPECTATIONS BY CONDITIONING 127 Therefore, conditioned on X Y = n, X is Binomial(n, λ1 λ1λ2 Example 112 Let T1,T2 be two independent, Exponential(λ) random variables, and let S1 = T1, S2 = T1 T2Compute fS1(s1S2 = s2) First,{(B/% * E`(@CN z &{fJ^!#EIEC!A N e B u ȃX k s A R Ă I y Ȃ Ⴄ A ȃt B M A 恙 ͉ Ă ́H X k s A N e B u ő ̂ˁI ȕ\ ߂ Ⴉ 킢 ނ ĐV o I I T C Y F } X R b g @ 2cm @ S @ 10cm A t B G C g N 쐬 X k s ` g уX g b v i u ɃI v J j y0222 t 5 z X k s ` g уX g b v i ݂ j y0222 t 5 z X k s ` g уX g b v i x _ E ݁j y0222 t 5 z X k s ` g уX g b v i t B b V O j y0222 t 5 z X k s ` g

315 f g h ` _ k l \ h l j m ^ h \ _ i h t e Z j k d Z b k l h j b y g Z g _ f k d b b t e Z j k d b _ a b d K i h j _ ^ ^ h k l h \ _ j g b ^ Z g g b _ i h qKa ls f pr u&euu d j jgs gS aA kS j ,d x q Lr kk h eS a v kS j d ¡x kA v kS j ok s ;Links with this icon indicate that you are leaving the CDC website The Centers for Disease Control and Prevention (CDC) cannot attest to the accuracy of a nonfederal website Linking to a nonfederal website does not constitute an endorsement by CDC or any of its employees of the sponsors or the information and products presented on the website

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By Jensen's inequality, Ef(X) ≥ f(EX) for any convex function f If f is twice differentiable and its second derivative is nonnegative, then f is convex For f(x) = xk, the second derivative is f00(x) = k(k −1)xk−2 which is nonnegative if x ≥ 0 2 (MU 27) Let X and Y be independent geometric random variables, where X hasThis list of all twoletter combinations includes 1352 (2 × 26 2) of the possible 2704 (52 2) combinations of upper and lower case from the modern core Latin alphabetA twoletter combination in bold means that the link links straight to a Wikipedia article (not a disambiguation page) As specified at WikipediaDisambiguation#Combining_terms_on_disambiguation_pages, terms which differ only in( v"q6 0 0" " b6 &,> 40 , #i£ }"q 6# 0 ro " )"