Exponential Expectation (a) Let X ∼ Exp(λ). Use induction to show that E[X k ] = k!/λ k for every k ∈ N. (b) For any |t| < λ, compute E[e tX] directly from the definition of expectation. (c) Using part (a), compute ∑ [infinity] k=0 E[X k ] k! t k . (d) Let M(t) = E[e tX] be a function defined for all t such that |t| < λ. What is dM(t) dt t=0 ? What is d 2M(t) dt2 t=0 ? How does each of these relate to the mean and variance of an Exp(λ) distribution?

Given: ab ≅ cd and ad ≅ bc prove: abcd is a parallelogram. statements reasons 1. ab ≅ cd; ad ≅ bc 1. given 2. ac ≅ ac 2. reflexive property 3. △adc ≅ △cba 3. ? 4. ∠dac ≅ ∠bca; ∠acd ≅ ∠cab 4. cpctc 5. ∠dac and ∠bca are alt. int. ∠s; ∠acd and ∠cab are alt. int. ∠s 5. definition of alternate interior angles 6. ab ∥ cd; ad ∥ bc 6. converse of the alternate interior angles theorem 7. abcd is a parallelogram 7. definition of parallelogram what is the missing reason in step 3? triangle angle sum theorem sas congruency theorem sss congruency theorem cpctc