Question

Consider a Poisson process on [0, \infty) with parameter \lambda and let T be a random variable independent of the process. Assume T has an exponentia

Discrete math
ANSWERED
asked 2021-05-05

Consider a Poisson process on \([0, \infty)\) with parameter \(\displaystyle\lambda\) and let T be a random variable independent of the process. Assume T has an exponential distribution with parameter v. Let \(N_{T}\) denote the number of particles in the interval \([0, T]\). Compute the discrete density of \(N_{T}\).

Answers (1)

2021-05-06

First of all, observe that \(N_T\) is a counter of particle in a random interval \([0,T].\) Therefore, \(N_T\) hs discrete support in \(N_0\). For \(k \in N_0\) we use law of the total probability to obtain that
\(\displaystyle{P}{\left({N}_{{T}}={k}\right)}={\int_{{{0}}}^{{\infty}}}{P}{\left({N}_{{T}}={k}{\mid}{T}={t}\right)}{{f}_{{T}}{\left({t}\right)}}{\left.{d}{t}\right.}={\int_{{{0}}}^{{\infty}}}{P}{\left({N}_{{t}}={k}{\mid}{T}={t}\right)}{{f}_{{T}}{\left({t}\right)}}{\left.{d}{t}\right.}\)
\(\displaystyle{\int_{{{0}}}^{{\infty}}}{P}{\left({N}_{{t}}={k}\right)}{{f}_{{T}}{\left({t}\right)}}{\left.{d}{t}\right.}={\int_{{{0}}}^{{\infty}}}{\frac{{{\left(\lambda{t}\right)}^{{{f}}}}}{{{k}!}}}{e}^{{-\lambda{t}}}\cdot{v}{e}^{{-{v}{t}}}{\left.{d}{t}\right.}={\frac{{\lambda^{{{k}}}\cdot{v}}}{{{k}!}}}{\int_{{{0}}}^{{\infty}}}{t}^{{{k}}}{e}^{{-{\left(\lambda+{v}\right)}^{{{t}}}}}{\left.{d}{t}\right.}\)
where in third equality we have used that T and considered Poisson process are independent. Now, let’s a calculate this integral. The key here is to use substitution \(\displaystyle{u}={\left(\lambda+{v}\right)}{t}\) and use properties of gamma function. We have that
\(\displaystyle{\int_{{{0}}}^{{\infty}}}{t}^{{{k}}}{e}^{{-{\left(\lambda+{v}\right)}{t}}}{\left.{d}{t}\right.}={\frac{{{1}}}{{{\left(\lambda+{v}\right)}^{{{k}+{1}}}}}}{\int_{{{0}}}^{{\infty}}}{u}^{{{k}}}{e}^{{-{u}}}{d}{u}={\frac{{Г{\left({k}+{1}\right)}}}{{{\left(\lambda+{v}\right)}^{{{k}+{1}}}}}}={\frac{{{k}!}}{{{\left(\lambda+{v}\right)}^{{{k}+{1}}}}}}.\) Finally, we have obtained that
\(\displaystyle{P}{\left({N}_{{T}}={k}\right)}={\frac{{\lambda^{{{k}}}\cdot{v}}}{{{k}!}}}\cdot{\frac{{{k}!}}{{{\left(\lambda+{v}\right)}^{{{k}+{1}}}}}}={\frac{{\lambda^{{{k}}}\cdot{v}}}{{{\left(\lambda+{v}\right)}^{{{k}+{1}}}}}}\)
\(\displaystyle{P}{\left({N}_{{T}}={k}\right)}={\frac{{\lambda^{{{k}}}\cdot{v}}}{{{\left(\lambda+{v}\right)}^{{{k}+{1}}}}}}\) for \(\displaystyle{k}\in{N}_{{0}}\)

0
 
Best answer

expert advice

Need a better answer?

Relevant Questions

asked 2021-07-02
An investor plans to put $50,000 in one of four investments. The return on each investment depends on whether next year’s economy is strong or weak. The following table summarizes the possible payoffs, in dollars, for the four investments.
Certificate of deposit
Office complex
Land speculation
Technical school
amp; Strong amp;6,000 amp;15,000 amp;33,000 amp;5,500
amp; Weak amp;6,000 amp;5,000 amp;−17,000 amp;10,000
Let V, W, X, and Y denote the payoffs for the certificate of deposit, office complex, land speculation, and technical school, respectively. Then V, W, X, and Y are random variables. Assume that next year’s economy has a 40% chance of being strong and a 60% chance of being weak. a. Find the probability distribution of each random variable V, W, X, and Y. b. Determine the expected value of each random variable. c. Which investment has the best expected payoff? the worst? d. Which investment would you select? Explain.
asked 2020-11-27

Assume a Poisson distribution with lambda \(= 5.0\). What is the probability that \(X = 1\)?

asked 2021-05-14
When σ is unknown and the sample size is \(\displaystyle{n}\geq{30}\), there are tow methods for computing confidence intervals for μμ. Method 1: Use the Student's t distribution with d.f. = n - 1. This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method. Method 2: When \(\displaystyle{n}\geq{30}\), use the sample standard deviation s as an estimate for σσ, and then use the standard normal distribution. This method is based on the fact that for large samples, s is a fairly good approximation for σσ. Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution. Consider a random sample of size n = 31, with sample mean x¯=45.2 and sample standard deviation s = 5.3. (c) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?
asked 2021-05-12
Let \(X_{1},X_{2},...,X_{6}\) be an i.i.d. random sample where each \(X_{i}\) is a continuous random variable with probability density function
\(f(x)=e^{-(x-0)}, x>0\)
Find the probability density function for \(X_{6}\).
asked 2021-06-06

Let X and Y be independent, continuous random variables with the same maginal probability density function, defined as
\(f_{X}(t)=f_{Y}(t)=\begin{cases}\frac{2}{t^{2}},\ t>2\\0,\ otherwise \end{cases}\)
(a)What is the joint probability density function f(x,y)?
(b)Find the probability density of W=XY. Hind: Determine the cdf of Z.

asked 2021-03-07
M. F. Driscoll and N. A. Weiss discussed the modeling and solution of problems concerning motel reservation networks in “An Application of Queuing Theory to Reservation Networks” (TIMS, Vol. 22, No. 5, pp. 540–546). They defined a Type 1 call to be a call from a motel’s computer terminal to the national reservation center. For a certain motel, the number, X, of Type 1 calls per hour has a Poisson distribution with parameter \(\displaystyle\lambda={1.7}\).
Determine the probability that the number of Type 1 calls made from this motel during a period of 1 hour will be:
a) exactly one.
b) at most two.
c) at least two.
(Hint: Use the complementation rule.)
d. Find and interpret the mean of the random variable X.
e. Determine the standard deviation of X.
asked 2021-05-26
Random variables \(X_{1},X_{2},...,X_{n}\) are independent and identically distributed. 0 is a parameter of their distribution.
If \(X_{1}, X_{2},...,X_{n}\) are Normally distributed with unknown mean 0 and standard deviation 1, then \(\overline{X} \sim N(\frac{0,1}{n})\). Use this result to obtain a pivotal function of X and 0.
asked 2021-05-09
The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus
\(\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}\)
where A is the cross-sectional area of the vehicle and \(\displaystyle{C}_{{d}}\) is called the coefficient of drag.
Part A:
Consider a vehicle moving with constant velocity \(\displaystyle\vec{{{v}}}\). Find the power dissipated by form drag.
Express your answer in terms of \(\displaystyle{C}_{{d}},{A},\) and speed v.
Part B:
A certain car has an engine that provides a maximum power \(\displaystyle{P}_{{0}}\). Suppose that the maximum speed of thee car, \(\displaystyle{v}_{{0}}\), is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power \(\displaystyle{P}_{{1}}\) is 10 percent greater than the original power (\(\displaystyle{P}_{{1}}={110}\%{P}_{{0}}\)).
Assume the following:
The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage, \(\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}\), is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.
asked 2021-06-04
Let \(X_{1}, X_{2},...,X_{n}\) be n independent random variables each with mean 100 and standard deviation 30. Let X be the sum of these random variables.
Find n such that \(Pr(X>2000)\geq 0.95\).
asked 2021-06-03

Compute the distribution of \(X+Y\) in the following cases:
X and Y are independent Poisson random variables with means respective \(\lambda_{1} and \lambda_{2}\).

...