Problem 1 Let X1, X2, X3, .., X, be a random sample from an exponential distribution with parameter 0, i.e., fx, (x;0) = 0e¯®*u(). Our goal is to find a (1 – a)100% confidence interval for 0. To do this, we need to remember a few facts about the gamma distribution. More specifically, If Y = X1 + X2 +.+ Xn, where the X;'s are independent Exponential(0) random variables, then Y ~ Gamma(n,0). Thus, the random variable Q defined as Q = 0(X1 + X2 +..+X„) has a Gamma(n, 1) distribution. Let us define yp,n as follows. For any pE [0,1] and ne N, we define Youn as the real value for which P(Q > Ypm) = P, where Q~ Gamma(n, 1). a. Explain why Q = 0(X1 + X2 + + Xn) is a pivotal quantity. b. Using Q and the definition of yp,n, construct a (1 - a)100% confidence interval for 0.
Problem 1 Let X1, X2, X3, .., X, be a random sample from an exponential distribution with parameter 0, i.e., fx, (x;0) = 0e¯®*u(). Our goal is to find a (1 – a)100% confidence interval for 0. To do this, we need to remember a few facts about the gamma distribution. More specifically, If Y = X1 + X2 +.+ Xn, where the X;'s are independent Exponential(0) random variables, then Y ~ Gamma(n,0). Thus, the random variable Q defined as Q = 0(X1 + X2 +..+X„) has a Gamma(n, 1) distribution. Let us define yp,n as follows. For any pE [0,1] and ne N, we define Youn as the real value for which P(Q > Ypm) = P, where Q~ Gamma(n, 1). a. Explain why Q = 0(X1 + X2 + + Xn) is a pivotal quantity. b. Using Q and the definition of yp,n, construct a (1 - a)100% confidence interval for 0.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![Problem 1
Let X1, X2, X3,.
..., X, be a random sample from an exponential distribution with
parameter 0, i.e.,
fx,(x; 0) = 0e¯®*u(x).
Our goal is to find a (1 – a)100% confidence interval for 0. To do this, we need to
remember a few facts about the gamma distribution. More specifically, If
Y = X1 + X2 + …+ Xn, where the X;'s are independent Exponential(8) random
variables, then Y ~ Gamma(n,0). Thus, the random variable Q defined as
Q = 0(X1 + X2 +..+X„)
has a Gamma(n, 1) distribution. Let us define yp,n as follows. For any p E (0, 1] and
n e N, we define Yp,n as the real value for which
P(Q > 7pm) = P,
where Q ~ Gamma(n, 1).
a. Explain why Q = 0(X1+ X2 + ..+ Xn) is a pivotal quantity.
b. Using Q and the definition of yp,n, construct a (1 - a)100% confidence interval for
0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1917885f-ce9d-477c-bc45-27ba1c10abf2%2F1eca171e-bbad-4d37-a73d-f246e6d8f21d%2Fcxcdprs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 1
Let X1, X2, X3,.
..., X, be a random sample from an exponential distribution with
parameter 0, i.e.,
fx,(x; 0) = 0e¯®*u(x).
Our goal is to find a (1 – a)100% confidence interval for 0. To do this, we need to
remember a few facts about the gamma distribution. More specifically, If
Y = X1 + X2 + …+ Xn, where the X;'s are independent Exponential(8) random
variables, then Y ~ Gamma(n,0). Thus, the random variable Q defined as
Q = 0(X1 + X2 +..+X„)
has a Gamma(n, 1) distribution. Let us define yp,n as follows. For any p E (0, 1] and
n e N, we define Yp,n as the real value for which
P(Q > 7pm) = P,
where Q ~ Gamma(n, 1).
a. Explain why Q = 0(X1+ X2 + ..+ Xn) is a pivotal quantity.
b. Using Q and the definition of yp,n, construct a (1 - a)100% confidence interval for
0.
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