The exponential distribution is a probability distribution for non-negative real numbers. It is often used to model waiting or survival times. The version that we will look at has a probability density function of the form p(y|v) = exp(-e-ºy-v) (1) where y ER+, i.e., y can take on the values of non-negative real numbers. In this form it has one parameter: a log-scale parameter v. If a random variable follows an exponential distribution with log-scale v we say that Y~ Exp(v). If Y~ Exp(v), then E [Y] =e" and V [Y] = e²º 1. Produce a plot of the exponential probability density function (1) for the values y € (0, 10), for v = 1, v = 0.5 and v= 2. Ensure the graph is readable, the axis are labeled appropriately and a legend is included. ta 2. Imagine we are given a sample of n observations y = (y₁,..., yn). Write down the joint proba- bility of this sample of data, under the assumption that it came from an exponential distribution with log-scale parameter v (i.e., write down the likelihood of this data). Make sure to simplify your expression, and provide working. (hint: remember that these samples are independent and identically distributed.). " 3. Take the negative logarithm of your likelihood expression and write down the negative log- likelihood of the data y under the exponential model with log-scale v. Simplify this expression. 1-4- 4. Derive the maximum likelihood estimator û for v. That is, find the value of v that minimises the negative log-likelihood. You must provide working.!! MA m. 5. Determine the approximate bias and variance of the maximum likelihood estimator û of v for the exponential distribution. (hints: utilise techniques from Lecture 2, Slide 27 and the mean and variance of the sample mean) wrival
The exponential distribution is a probability distribution for non-negative real numbers. It is often used to model waiting or survival times. The version that we will look at has a probability density function of the form p(y|v) = exp(-e-ºy-v) (1) where y ER+, i.e., y can take on the values of non-negative real numbers. In this form it has one parameter: a log-scale parameter v. If a random variable follows an exponential distribution with log-scale v we say that Y~ Exp(v). If Y~ Exp(v), then E [Y] =e" and V [Y] = e²º 1. Produce a plot of the exponential probability density function (1) for the values y € (0, 10), for v = 1, v = 0.5 and v= 2. Ensure the graph is readable, the axis are labeled appropriately and a legend is included. ta 2. Imagine we are given a sample of n observations y = (y₁,..., yn). Write down the joint proba- bility of this sample of data, under the assumption that it came from an exponential distribution with log-scale parameter v (i.e., write down the likelihood of this data). Make sure to simplify your expression, and provide working. (hint: remember that these samples are independent and identically distributed.). " 3. Take the negative logarithm of your likelihood expression and write down the negative log- likelihood of the data y under the exponential model with log-scale v. Simplify this expression. 1-4- 4. Derive the maximum likelihood estimator û for v. That is, find the value of v that minimises the negative log-likelihood. You must provide working.!! MA m. 5. Determine the approximate bias and variance of the maximum likelihood estimator û of v for the exponential distribution. (hints: utilise techniques from Lecture 2, Slide 27 and the mean and variance of the sample mean) wrival
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...
Related questions
Question
![The exponential distribution is a probability distribution for non-negative real numbers. It is often
used to model waiting or survival times. The version that we will look at has a probability density
function of the form
p(y|v) = exp(-e¯ºy — v)
(1)
where y R+, i.e., y can take on the values of non-negative real numbers. In this form it has one
parameter: a log-scale parameter v. If a random variable follows an exponential distribution with
log-scale v we say that Y~ Exp(v). If Y~ Exp(v), then E [Y] =e" and V [Y] = e²v.
1. Produce a plot of the exponential probability density function (1) for the values y E (0, 10), for
v = 1, v = 0.5 and v= 2. Ensure the graph is readable, the axis are labeled appropriately and
a legend is included.
ta
il
2. Imagine we are given a sample of n observations y = (y₁,..., yn). Write down the joint proba-
bility of this sample of data, under the assumption that it came from an exponential distribution
with log-scale parameter v (i.e., write down the likelihood of this data). Make sure to simplify
your expression, and provide working. (hint: remember that these samples are independent and
identically distributed.)
j
3. Take the negative logarithm of your likelihood expression and write down the negative log-
likelihood of the data y under the exponential model with log-scale v. Simplify this expression.
IH
4. Derive the maximum likelihood estimator û for v. That is, find the value of v that minimises the
negative log-likelihood. You must provide working.!! VA
m...
5. Determine the approximate bias and variance of the maximum likelihood estimator û of v for the
exponential distribution. (hints: utilise techniques from Lecture 2, Slide 27 and the mean and
variance of the sample mean) [w.kz]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d711242-a88c-4a58-a190-33258ba50e1c%2Fdd86b801-ecd9-468d-a602-41be4261edb8%2Fe747oxp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The exponential distribution is a probability distribution for non-negative real numbers. It is often
used to model waiting or survival times. The version that we will look at has a probability density
function of the form
p(y|v) = exp(-e¯ºy — v)
(1)
where y R+, i.e., y can take on the values of non-negative real numbers. In this form it has one
parameter: a log-scale parameter v. If a random variable follows an exponential distribution with
log-scale v we say that Y~ Exp(v). If Y~ Exp(v), then E [Y] =e" and V [Y] = e²v.
1. Produce a plot of the exponential probability density function (1) for the values y E (0, 10), for
v = 1, v = 0.5 and v= 2. Ensure the graph is readable, the axis are labeled appropriately and
a legend is included.
ta
il
2. Imagine we are given a sample of n observations y = (y₁,..., yn). Write down the joint proba-
bility of this sample of data, under the assumption that it came from an exponential distribution
with log-scale parameter v (i.e., write down the likelihood of this data). Make sure to simplify
your expression, and provide working. (hint: remember that these samples are independent and
identically distributed.)
j
3. Take the negative logarithm of your likelihood expression and write down the negative log-
likelihood of the data y under the exponential model with log-scale v. Simplify this expression.
IH
4. Derive the maximum likelihood estimator û for v. That is, find the value of v that minimises the
negative log-likelihood. You must provide working.!! VA
m...
5. Determine the approximate bias and variance of the maximum likelihood estimator û of v for the
exponential distribution. (hints: utilise techniques from Lecture 2, Slide 27 and the mean and
variance of the sample mean) [w.kz]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 15 images
Recommended textbooks for you
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON