Exponential distribution is a special case of gamma distribution.Review LaterTrueFalse

When a certain procedure consists of n𝑛 independent steps and each step takes Exponential(λ𝜆) amount of time then the total time for these n𝑛 steps will follow:a.Exponential distributionb.Gamma distributionc.Normal distributiond.Uniform distribution

Explain exponential distribution. Explain its relationship with Poisson distribution.

A sample of a radioactive material is studied in a lab. There are 1053 gamma ray emissions over 390 seconds.Use the Poisson distribution to find the probability that exactly 2 gamma rays are emitted in a single second.Do not round intermediate computations, and round your answer to three decimal places.(If necessary, consult a list of formulas.)

Define a Gamma function

Question 3. [9 Marks]. Recall that the exponential PDF has the form f (t; λ) = λe−λtwhere t, λ > 0 and λ is known as the rate parameter. Consider a job shop that consists of3 identical machines and 2 technicians. Suppose that, the amount of time each machineoperates before breaking down is exponentially distributed with rate parameter 0.1, and theamount of time a technician takes to fix a machine is exponentially distributed with rateparameter 0.4. Suppose that all the times to breakdown and times to repair are independentrandom variables and let X(t) be the number of machines which are operating at time t.(a) Determine the Q-matrix for this Markov process.(b) Write the forward equations involving the P ′0j (t), j = 0, 1, 2, 3, in terms ofpj (t) := P0j (t) = P(X(t) = j|X(0) = 0).(c) Obtain the equilibrium probabilities pj = limt→∞ pj (t).(d) What is the average number of busy technicians in the long-run? [Note: do not expectyour answer to be an integer.]