(a) Radioactive particles are detected by a counter according to a Poisson process with rate parameter λ = 0.5 particles per second. What is the probability that two particles are detected in any given one second interval ? (b) Page visits to a particular website occur according to a Poisson process with rate parameter λ = 20 per minute. What is the expected number of visits to the website in any given one hour period ? (c) What is the probability that the time of the first event that is observed to occur in a Poisson process with rate λ per unit time, after initiation at t = 0, occurs later than time t = t0, for fixed value t0 ? Justify your answer. The Poisson process is a model for events that occur in continuous time, at a constant rate ) > 0 perunit time, with events occurring independently of each other. Specifically, if X (t) is the discreterandom variable recording the number of events that are observed to occur in the interval [0, t),then we have that X (t) ~ Poisson(At), that isp(x) = P(X(t) = x) = e¬At (At)“x!x = 0, 1, 2, ...and zero otherwise. Also, the counts of events in disjoint time intervals are probabilisticallyindependent: for example, for intervals [0, t) and [t, t'+ s), the numbers of events in the twointervals, X1 and X2 say, have the propertyP(X1 = #10X2 = x2) = P(X1 = #1)P(X2 = x2)%3DwithX1 ~ Poisson(At)X2 ~ Poisson(As).With this information, answer the following questions based on the Poisson process model andits relationship with the Poisson distribution.

As per our guidelines, we are allowed to answer first question only. Thanks (a) Radioactive…

Math

Probability

(a) Radioactive particles are detected by a counter according to a Poisson process with rate parameter λ = 0.5 particles per second. What is the probability that two particles are detected in any given one second interval ?

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Similar questions

A day's sales in $1000 units at a gas station have a gamma distribution with parameters k = 5 and 1 = 0.9. Then DETERMINE (i) The expectation of a day's sales? (ii) The standard deviation of a day's sales? (iii) The upper and lower quartiles of a day's sales? (iv) The probability that a day's sales are more than $6000?
3) (4 points) Consider a population with the mean u = 51, 000 and the standard deviation o = 5, 000. For a random sample of size n = 100, find the probability that the sample mean X will be between 50, 000 and 51, 500.
What is the probability that more than 5 cars arrive within a one hour period? What is P(X>5)?
The molecular weight of a particular polymer should fall between 750 and 1,035. Thirty samples of this material were analyzed with the results X-bar = 902 and s = 90. Assume that molecular weight is normally distributed.
REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults†. Assume that REM sleep time is normally distributed for both children and adults. A random sample of n1 = 9 children (9 years old) showed that they had an average REM sleep time of x1 = 2.5 hours per night. From previous studies, it is known that ?1 = 0.6 hour. Another random sample of n2 = 9 adults showed that they had an average REM sleep time of x2 = 2.10 hours per night. Previous studies show that ?2 = 0.5 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 10% level of significance. Solve the problem using both the traditional method and the P-value method. (Test the difference ?1 − ?2. Round the test statistic and critical value to two decimal places. Round the P-value to four decimal places.) test statistic critical value…
Suppose the random variable, X, follows a geometric distribution with parameter (0< 0 <1). Let X₁, X₂, X, be a random sample of size n from the population of X. (d) Find the method of moments estimator (mme) of 0. (e) Show how the mle of 0 is related to its mme. Explain if it is always true. (f) Verify if X, is a sufficient statistic for 8 or not. (g) Justify if = is an unbiased estimator for 6.
Weights of Lamb after eight weeks after born are normally distributed with mean µ = 14 kg and variance σ 2 = 4. a)Find the probability p that a single lamb has weight less than 12 kg. b) Find the probability that exactly 3 of the next 5 lambs examined will have weights less than 12 kg.
What type of distribution is appropriate to model the time between two consecutive departures from the gate of an airport Poisson Binomial Exponential Geometric
REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults†. Assume that REM sleep time is normally distributed for both children and adults. A random sample of n1 = 8 children (9 years old) showed that they had an average REM sleep time of x1 = 2.5 hours per night. From previous studies, it is known that ?1 = 0.9 hour. Another random sample of n2 = 8 adults showed that they had an average REM sleep time of x2 = 1.60 hours per night. Previous studies show that ?2 = 0.6 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance. what is the sample test statistic find or estimate p value
The number of errors per 1000 words made by a particular typist has a Poisson distribution with a mean of 2.5.(a) Find the probability that, in a 4000 word script, there should at least be 14 typing errors.(b) Using a suitable approximation, show that the probability will be more than 25 errors in a 10000 word dissertation.
Question 7 According to the data, the weight of a randomly selected checked-in luggage has a normal distribution with a mean of 51 lbs and a standard deviation of 9 lbs. Let X be the weight of a randomly selected checked-in luggage and let S be the total weight of a random sample of size 33. 1. Describe the probability distribution of X and state its parameters and o: X~ Select an answer (μ = Select an answer unknown X² B N T 2. Use the Central Limit Theorem and find the probability that the weight of a randomly selected checked-in luggage is less than 61 lbs. (Round the answer to 4 decimal places) Check-In Select an answer Select an answer the original population is normally distributed the sample size is large (n>30) although the distribution of the original population is unknown the sample size is small (n<30) and the distribution of the original population is unknown the distribution of the original population is unknown S~ Select an answer (s= Select an answer σ= to describe the…
The arrival of customers to a small local food restaurant may be modeled by a Poisson process with rate of 1 per 15 min period. Assume customer will only arrive if there is at least one customer present. Each customer on average stays for a time exponentially distributed with mean 20 minutes. This is modeled as a birth-death process. (i) Compute the probability that the capacity of 20 will be reached if we start with 5 customers.