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Poisson Random Variables Part 1 A process is modeled by a random variable with density Poisson(k;...
7. A random process may be modeled as a random variable with the probability density function...
7. A random process may be modeled as a random variable with the probability density function fx(x) fx(x) fx(x) x < 1/2 x 1 /2 = 2-4x = 4x-2 = 0 (a) Show that Jdfx(r)dr = 1 (b) Find the CDF, Fx() (c) Find the probability that X is between 1/3 and 2/3. (d) Find the expected value of X. (e) Find the mode of X (f) Find the median of X. (g) Find the variance of X and the...
PROBLEM 2 The number of accidents in a certain city is modeled by a Poisson random variable with average rate of 10 acc...
PROBLEM 2 The number of accidents in a certain city is modeled by a Poisson random variable with average rate of 10 accidents per day. Suppose that the number of accidents in different days are independent. Use the central limit theorem to find the probability that there will be more than 3800 accidents in a certain year. Assume that there are 365 days in a year.
4-78. Suppose that X is a Poisson random variable with λ 6. (a) Compute the exact...
4-78. Suppose that X is a Poisson random variable with λ 6. (a) Compute the exact probability that X is less than 4. (b) Approximate the probability that X is less than 4 and com- pare to the result in part (a). (c) Approximate the probability that 9 < X <12
P7 continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The...
P7
continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY...
Exercise 3. We say that a family of random variables (X)teo.) converges in probability to a random variable X (notation...
Exercise 3. We say that a family of random variables (X)teo.) converges in probability to a random variable X (notation: Xt-, X) if for everye> 0, liml+xPfXt-X) > e) = 0. Suppose that (Nt)t20 is a Poisson process of rate X. Show that Nt/t P+ λ. This shows that the rate measures the average frequency or density of arrivals. What about N1 λ?
Exercise 3. We say that a family of random variables (X)teo.) converges in probability to a random...
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability...
Recall that a discrete random variable X has Poisson
distribution with parameter λ if the probability mass function of
X
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is r E 0,1,2,...) This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a) Prove by direct computation that the mean of...
Suppose that visits to a website can be modeled by a Poisson process with a rate λ=10 per hour (a...
suppose that visits to a website can be modeled by a Poisson
process with a rate λ=10 per hour
(a) What is the probability that there are more than or equal to
2 visits within a given 1/2 hour interval
(b) A supervisor starts to monitor the website from the start of
a new shift. then what is the expected value of time waited by the
supervisor until the 10th visit to the website during that
shift?
Suppose that visits...
Problem 1: 10 points Assume that a random variable X follows the Poisson distribution with intensity-A,...
Problem 1: 10 points Assume that a random variable X follows the Poisson distribution with intensity-A, that is k! for k 0,1,2, . Using the identity (valid for all real t) exp(t) = Σ冠. k! k=0 derive the probability that X takes an even value, that is PIX is even
The number of fish that a fisherman catches in a day is a Poisson random variable...
The number of fish that a fisherman catches in a day is a Poisson random variable with mean = 30. However, on average, the fisherman throws back two out of every three fish he catches. (a) What is the probability that, on a given day, the fisherman takes home n fish. (b) What is the mean and variance of the number of fish he catches (c) What is the mean and variance of the number of fish he takes home...
5. Let X1, X2,... , X100 be i.i.d. random variables, following the normal distribution N(0, 102)....
5. Let X1, X2,... , X100 be i.i.d. random variables, following the normal distribution N(0, 102). Let α denote the probability that there are at least 3 variables among them whose absolute value is larger than 19.6. Compute α, and give an approxi- mate value of α with an error less than 0.01 according to the Poisson distribution. 15pts]
5. Let X1, X2,... , X100 be i.i.d. random variables, following the normal distribution N(0, 102). Let α denote the probability...
7. A random process may be modeled as a random variable with the probability density function fx(x) fx(x) fx(x) x < 1/2 x 1 /2 = 2-4x = 4x-2 = 0 (a) Show that Jdfx(r)dr = 1 (b) Find the CDF, Fx() (c) Find the probability that X is between 1/3 and 2/3. (d) Find the expected value of X. (e) Find the mode of X (f) Find the median of X. (g) Find the variance of X and the...
PROBLEM 2 The number of accidents in a certain city is modeled by a Poisson random variable with average rate of 10 accidents per day. Suppose that the number of accidents in different days are independent. Use the central limit theorem to find the probability that there will be more than 3800 accidents in a certain year. Assume that there are 365 days in a year.
4-78. Suppose that X is a Poisson random variable with λ 6. (a) Compute the exact probability that X is less than 4. (b) Approximate the probability that X is less than 4 and com- pare to the result in part (a). (c) Approximate the probability that 9 < X <12
P7
continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY...
Exercise 3. We say that a family of random variables (X)teo.) converges in probability to a random variable X (notation: Xt-, X) if for everye> 0, liml+xPfXt-X) > e) = 0. Suppose that (Nt)t20 is a Poisson process of rate X. Show that Nt/t P+ λ. This shows that the rate measures the average frequency or density of arrivals. What about N1 λ?
Exercise 3. We say that a family of random variables (X)teo.) converges in probability to a random...
Recall that a discrete random variable X has Poisson
distribution with parameter λ if the probability mass function of
X
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is r E 0,1,2,...) This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a) Prove by direct computation that the mean of...
suppose that visits to a website can be modeled by a Poisson
process with a rate λ=10 per hour
(a) What is the probability that there are more than or equal to
2 visits within a given 1/2 hour interval
(b) A supervisor starts to monitor the website from the start of
a new shift. then what is the expected value of time waited by the
supervisor until the 10th visit to the website during that
shift?
Suppose that visits...
Problem 1: 10 points Assume that a random variable X follows the Poisson distribution with intensity-A, that is k! for k 0,1,2, . Using the identity (valid for all real t) exp(t) = Σ冠. k! k=0 derive the probability that X takes an even value, that is PIX is even
5. Let X1, X2,... , X100 be i.i.d. random variables, following the normal distribution N(0, 102). Let α denote the probability that there are at least 3 variables among them whose absolute value is larger than 19.6. Compute α, and give an approxi- mate value of α with an error less than 0.01 according to the Poisson distribution. 15pts]
5. Let X1, X2,... , X100 be i.i.d. random variables, following the normal distribution N(0, 102). Let α denote the probability...
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