Consider a random graph G(N, p) with 2-N² √N + 3N Therefore the random graph has not has a giant component in the limit N → ∞. P = 2- In the limit N → ∞ the average degree (k) is given by ∞00 2 None of the above

Could you solve (d),(e),(f)? Thank you.

(b) Describe the sigma-field F2. (Write down All elements of F2). Hint: How many outcomes if we toss a coin two times? Then how many subsets for the set including all of the outcomes? Given {N(t)} is a Poisson process, please calculate (a) P(N(4)=2|N(2)=1) (b) E[N(4)|N(2)=1] (c) Var[N(4)|N(2)=1] (d) P(N(4)=2|N(10)=5), also calculate E[N(4)|N(10)=5] and Var[N(4)|N(10)=5]

Using the joint distribution describing X and Y, the following formula was derived E(XKym) = 2km+3(m+k)+4 (k+1)(k+2)(m+1)(m+2) where k and m are both nonnegative constants. Simplify Var( 2X - 3Y) [ Select ] - Var(X) [Select] [ Select ] Var(Y) [Select] [ Select ] Cov(X,Y)

Consider a Poisson process of intensity λ > 0 events per hour. Suppose that exactly one event occurs in the first hour. Let X be the time at which this event occurs, measured in hours (so that the possible values of X are the reals in the interval [0, 1]). (a)  What is P (0 ≤ X ≤ a) for a ∈ [0, 1]? (Of course, we are conditioning on exactly one event occurring in the first hour, as described.) Hint: Consider the number of events that occur in [0, a] and in (a, 1], similar to question #5 on the previous homework. (b)  What is the distribution of X, under the same conditions? (You might want to wait until Monday’s class after break to do this last part.)

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Consider a random graph G(N, p) with .In the limit N → ∞ the average degree (k) is given by 0 0 0 0 2/3 ○ None of the above Therefore the random graph O has not O has a giant component in the limit N → ∞. P = e² In 2 3N3/2

Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. 35. v aniwollo 36. sin x?

Suppose X₁, X2,... are i.i.d. with moment generating function Mx(t). Let N be a nonnegative integer-valued random variable, independent of the X₁, with moment generating function My(t). Consider the random sum S X₁. In terms of Mx and MN, find the moment generating function of S. Use this to rederive the mean of S. Specializing to the case where the X, are Poisson with mean λ and N is Poisson with mean μ, what is the moment generating function of S?

For the Normal-Cauchy Bayes estimator: 8(x) = 5⁰⁰ ง -e-(x-9) ²/² dv -∞0⁰ 1 +92 e 1 ∞ 1 +9² e−(x−v)² dv Build an Accept-Reject algorithm based on the Cauchy candidate to generate a sample from the posterior distribution and then deduce the estimator.