Reference no: EM132489375

Question 1. Assume we roll a fair four-sided die marked with 1, 2, 3 and 4.
(a) Find the probability that the outcome 1 is first observed after 5 rolls.
(b) Find the expected number of rolls until outcomes 1 and 2 are both observed.
(c) Find the expected number of rolls until the outcome 3 is observed three times.
(d) Find the probability that the outcome 3 is observed exactly three times in 10 rolls given that it is first observed after 5 rolls.
(e) Find the probability that the outcome 3 is first observed after 5 rolls given that it is observed exactly three times in 10 rolls

Question 2. The traffic incidents in Melbourne and Sydney follow a Poisson process with the rate of 5 and 6 incidents per hour, respectively.
(a) Find the probability that no traffic accidents will occur in Melbourne in the next 30 minutes.
(b) Find the expected time (in minutes) until 10 new incidents occur in Sydney.
(c) Find the expected time (in minutes) until 10 new incidents in total occur in Melbourne and Sydney. Hint: For this and the next question, you can assume that traffic incidents in the two cities are independent.
(d) Assuming exactly 5 traffic incidents in total will occur in Melbourne and Sydney in the next 10 minutes, find the probability that not more than 2 traffic incidents will occur in Melbourne during the same period of time. Hint: Use the formula of conditional probability.

Question 3. Let X has the probability density function (pdf)
fx(x)=  C₁, if 0 < x ≤ 1,
fx(x)= C₂x, if 1 < x ≤ 4,
fx(x)= 0, otherwise.

Assume that the mean E(X) = 2.57.
(a) Find the normalizing constants C₁ and C₂.
(b) Find the cdf of X, F_x.
(c) Find the variance Var(X) and the 0.28 quantile q0.2s of X.
(d) Let Y = kX². Find all constants k such that Pr(1 < Y < 9) = 0.035.

Hint: express the event {1 < Y < 9} in terms of the random variable X and then use the cdf of X, Fx.