Let X be a continuous random variable taking values in [a, b] with c.d.f. Fx which is strictly increasing on [a, b]. (a) Show that the random variable Fx (X) has a uniform distribution on [0, 1]. (b) Let U be a uniform random variable on [0, 1]. What is the distribution of the random variable Fx¹(U), where Fx¹ is the inverse of Fx? (c) Suppose that U₁, U2, ..., Un are a set of computer-generated pseudo-random numbers (assumed to be drawn from a uniform distribution on [0, 1]). How would you use them to simulate a random sample X₁, X2, ..., Xn from the distribution with density f(x) = μe-μα, x ≥ 0?
Let X be a continuous random variable taking values in [a, b] with c.d.f. Fx which is strictly increasing on [a, b]. (a) Show that the random variable Fx (X) has a uniform distribution on [0, 1]. (b) Let U be a uniform random variable on [0, 1]. What is the distribution of the random variable Fx¹(U), where Fx¹ is the inverse of Fx? (c) Suppose that U₁, U2, ..., Un are a set of computer-generated pseudo-random numbers (assumed to be drawn from a uniform distribution on [0, 1]). How would you use them to simulate a random sample X₁, X2, ..., Xn from the distribution with density f(x) = μe-μα, x ≥ 0?
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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Chapter1: Starting With Matlab
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Problem 1P
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Question
![Question 1
Let X be a continuous random variable taking values in [a, b] with c.d.f. Fx which is
strictly increasing on [a, b].
(a) Show that the random variable Fx(X) has a uniform distribution on [0, 1].
(b) Let U be a uniform random variable on [0, 1]. What is the distribution of the random
variable F¹(U), where Fx¹ is the inverse of Fx?
(c) Suppose that U1, U2, . Un are a set of computer-generated pseudo-random numbers
(assumed to be drawn from a uniform distribution on [0, 1]). How would you use them
to simulate a random sample X₁, X2,..., Xn from the distribution with density
f(x) = = με-μα,
x ≥ 0?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb924e824-f7b8-486a-bee8-48b1feb67b77%2Ffea20e82-2cc0-4c85-9596-43a66c1d385e%2Fpuaql38_processed.png&w=3840&q=75)
Transcribed Image Text:Question 1
Let X be a continuous random variable taking values in [a, b] with c.d.f. Fx which is
strictly increasing on [a, b].
(a) Show that the random variable Fx(X) has a uniform distribution on [0, 1].
(b) Let U be a uniform random variable on [0, 1]. What is the distribution of the random
variable F¹(U), where Fx¹ is the inverse of Fx?
(c) Suppose that U1, U2, . Un are a set of computer-generated pseudo-random numbers
(assumed to be drawn from a uniform distribution on [0, 1]). How would you use them
to simulate a random sample X₁, X2,..., Xn from the distribution with density
f(x) = = με-μα,
x ≥ 0?
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