Introduction to Probability Models and Applications 1st. Edition by N. Balakrishnan, Markos V. Koutras, Konstadinos G. Politis | 9781118123348 | 1118123344 | Holooly

Statistical Mechanics

Introduction to Probability Models and Applications

102 SOLVED PROBLEMS

Question: A.2

Consider the sum S = a + a𝑤 + a𝑤² +···+ a𝑤^n−1 = ∑i=1^n a𝑤^i−1 of the first n terms in a geometric series. Show that the sum S in the above formula equals ∑i=1^n a𝑤^i−1 = a(1 − 𝑤^n)/1 − 𝑤, provided that 𝑤 ≠ 1. (For 𝑤 = 1, we have trivially S = na.) ...

Verified Answer:

Multiplying each term of S by 𝑤 we obtain that [la...

Question: 7.15

The maximum time allowed for an exam is three hours. The time required, X, as a proportion of this maximum duration by a student to complete the exam (that is, if a student completes the exam in t hours, X takes the value t∕3) has a Beta distribution with parameters 𝛼 = 5 and 𝛽 = 2. (i) Find the ...

Verified Answer:

The density function of X is given by

f(x)=...

Question: 7.9

Suppose we want to estimate the proportion p (or equivalently, the percentage 100p%) of the persons who intend to vote for a certain political party in the forthcoming general elections. For this reason, we plan to take a sample of size n of voters and ask them about their intention (assuming they ...

Verified Answer:

This is a sampling problem. We have seen problems ...

Question: 7.7

An automatic machine is used to dispense a particular soft drink in bottles with a nominal quantity (volume) of 1.5l. If the machine puts more than 1.6l of the soft drink in a bottle, the excess quantity is wasted. The actual volume that the machine dispenses in the bottles is a random variable ...

Verified Answer:

Let X denote the random variable corresponding to ...

Question: 7.6

The time that John needs to get from his house to the University every morning is a continuous random variable which is assumed to follow the normal distribution with mean 𝜇 = 35 minutes and a standard deviation 𝜎 = 5 minutes.(i) Find the probability that on a particular day his journey takes (a) ...

Verified Answer:

Let X be the time that John’s journey takes on tha...

Question: 7.11

The lifetime X of a device is assumed to follow an exponential distribution with parameter 𝜆 > 0. (i) Calculate the probability that the lifelength of the device (a) exceeds its mean value; (b) is at least twice as much as its mean value. (ii) Find the length of time t0, such that the probability ...

Verified Answer:

Let F denote the distribution function of the life...

Question: 7.4

(Quantiles of the standard normal distribution) We have just seen how we can use the table of values for the standard normal distribution to find, for a given real z, the corresponding probability Φ(z) = P(Z ≤ z). In many cases, interest lies in the opposite direction, so that for a given value 𝛼 ...

Verified Answer:

Since for any real z

P(Z \gt z) = 1 − P(Z \...

Question: 7.2

Let X have a uniform distribution in the interval [𝛼, 𝛽]. What is the distribution of the linear transformation Y = 𝛾X + 𝛿 for 𝛾 > 0 and 𝛿 ∈ ℝ? ...

Verified Answer:

It is easier to work with distribution functions r...

Question: 7.1

Simon goes to his University class every weekday by using a train that leaves at 8:00 a.m. We assume that the duration of the journey, in minutes, is a random variable following the uniform distribution in the interval [58, 63]. Further, suppose that from the train platform he needs a 15-minute ...

Verified Answer:

Since the only source of uncertainty is the durati...

Question: 7.12

The time between two successive emails arriving at Julia’s email account follows the exponential distribution with a mean value of 90 minutes. (i) What is the probability that the time between two successive emails exceeds one hour? (ii) Given that 90 minutes have passed since Julia received her ...

Verified Answer:

Let us choose one hour as the time unit.³ Then, if...

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