The number of visits to a website is known to have a Poisson distribution with a mean of 10 visits per minute.

(a) What is the probability distribution for x, the number of visits per minute? p(x) =       

(b) What is the probability that the number of visits per minute is less than or equal to 11? (Round your answer to three decimal places.)

(c) What is the probability that the number of visits per minute is greater than 15? (Round your answer to three decimal places.)

(d) Within what limits does Tchebysheff's Theorem suggest you would expect the number of visits to this website to lie at least 89% of the time? (Round your answer up to the nearest whole number.) 0 to  visits

3. Consider any Random Walk W = (Wn)n≥0 on Z, that is, Wn := a + X1 + · · · + Xn whereX1,X2,... areIIDRVs.
Let r := P0(return to 0) be the probability that W returns to 0 given it starts at 0. Let N be the total number of visits to 0 including the visit at time 0. Since intuitively the Random Walk ‘starts afresh’ whenever it first returns to 0, show that for r ∈ [0, 1),

P0(N =k)=rk−1(1−r) (k=1,2,...),

so that the total number of visits to 0 when started at 0 is a Geometrically dis- tributed RV. Also note that P0(N = +∞) = 1 when r = 1.
Hence, calculate the expected total number of visits to 0 when starting at 0, E0(N), in terms of r.

On the other hand, with ξn := I{Wn=0}, note N = ξ0 +ξ1 +ξ2 +... where the sum counts 1 for every time 0 is visited. By taking expectations and comparing expressions, deduce that

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