24 Let X₁, X₂... X be a random sample of size n from a population with a probability density function f(x) and the corresponding umulative distribution function F(x). Let Y₁ < Y₂ < Y3 <. Yn be an order statistic of the random sample, where Y₁ is the smallest and n is the largest and Y is kth largest of X₁, X₂, ... Xn. 24(a) Show that the cumulative distribution functions of the random variables Y₁ and Yn are given by Fy₁ (3₁) = 1 [1 - F(y₁)]", Fy₁ = [F(yn)]". 24(b) Suppose the population density function is given by f(x) = 2x, 0 < x < 1, otherwise. how that the density functions of 9₁ (3₁) of Y₁ and gn (yn) of Yn are (using part(a)) 91 (3₁) = 2ny₁ (1-y)-¹, 0< y₁ <1, In (Yn) = 2ny2n-1, 24(c) The joint probability density function of Y₁ < Y₂ < Y3 <. Yn is given by 9(y₁, y2,, Yn) = n! f(y₁) f(y₂) f(yn), Y₁Y2 < < Yn. Suppose n = 3, write down the joint density g(y₁, 92, 93) of Y₁ < Y₂ < Y3 when 2x, 0
24 Let X₁, X₂... X be a random sample of size n from a population with a probability density function f(x) and the corresponding umulative distribution function F(x). Let Y₁ < Y₂ < Y3 <. Yn be an order statistic of the random sample, where Y₁ is the smallest and n is the largest and Y is kth largest of X₁, X₂, ... Xn. 24(a) Show that the cumulative distribution functions of the random variables Y₁ and Yn are given by Fy₁ (3₁) = 1 [1 - F(y₁)]", Fy₁ = [F(yn)]". 24(b) Suppose the population density function is given by f(x) = 2x, 0 < x < 1, otherwise. how that the density functions of 9₁ (3₁) of Y₁ and gn (yn) of Yn are (using part(a)) 91 (3₁) = 2ny₁ (1-y)-¹, 0< y₁ <1, In (Yn) = 2ny2n-1, 24(c) The joint probability density function of Y₁ < Y₂ < Y3 <. Yn is given by 9(y₁, y2,, Yn) = n! f(y₁) f(y₂) f(yn), Y₁Y2 < < Yn. Suppose n = 3, write down the joint density g(y₁, 92, 93) of Y₁ < Y₂ < Y3 when 2x, 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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could you solve last 3 questions?
Thank you
![Q4 Let X₁, X₂, ..., Xn be a random sample of size n from a population with a probability density function f(x) and the corresponding
cumulative distribution function F(x). Let Y₁ < Y₂ < Y3 < ... Yn be an order statistic of the random sample, where Y₁ is the smallest and
Yn is the largest and Yk is kth largest of X₁, X₂, ... Xn.
Q4(a) Show that the cumulative distribution functions of the random variables Y₁ and Y₁ are given by
Fy, (91₁) =1[1- F(y)]", FY₁ = [F(yn)]".
Q4(b) Suppose the population density function is given by
2x,
0
Show that the density functions of g₁ (₁) of Y₁ and 9n (yn) of Ynare (using part(a))
f(x) =
91 (9₁) = 2ny₁ (1 - y²)¹-¹, 0<y₁ <1, In (Yn) = 2ny2n-1, 0<yn <1.
Q4(c) The joint probability density function of Y₁ < Y₂ < Y3 <... Yn is given by
0 < x < 1,
otherwise.
g(y₁, Y2,..., Yn) = n!f(y₁)ƒ(Y2) ··· ƒ(Yn), Y₁ <Y2 <··· < Yn-
Suppose n = 3, write down the joint density g(y₁, 92, 93) of Y₁ < Y₂ < Y3 when
0 < x < 1,
otherwise.
91,n (y1, yn) =
f(x) =
2x,
0
. Now derive the marginal probability density of g₁ (3₁) of Y₁ and 93 (43) of Y3 and verify that it matches with the answer in part(b) for n = 3.
Q4(d) Note that the joint probability density function of (Y₁, Yn) is given for the special case of f(x) by
n!
¡ (2y₁)[y? — y?]”¯²(2yn), 0<Y₁ <Yn <1.
(n − 2)!
Write down the joint density function of 9₁,3(y₁, y3) of Y₁, Y3 when n = 3. Let Z₁ = Y - Y₁ and Z2 = Y3. Find the joint density of
(Z₁, Z₂) using the transformation method and deduce the probability density function of Z₁ which is the range of the random sample X₁,
X₂, and X3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fafa0dae5-21eb-4e3b-b875-4122619f3067%2F1fc84049-e4be-48e9-be7f-af704b8e34a5%2Fuheam5_processed.png&w=3840&q=75)
Transcribed Image Text:Q4 Let X₁, X₂, ..., Xn be a random sample of size n from a population with a probability density function f(x) and the corresponding
cumulative distribution function F(x). Let Y₁ < Y₂ < Y3 < ... Yn be an order statistic of the random sample, where Y₁ is the smallest and
Yn is the largest and Yk is kth largest of X₁, X₂, ... Xn.
Q4(a) Show that the cumulative distribution functions of the random variables Y₁ and Y₁ are given by
Fy, (91₁) =1[1- F(y)]", FY₁ = [F(yn)]".
Q4(b) Suppose the population density function is given by
2x,
0
Show that the density functions of g₁ (₁) of Y₁ and 9n (yn) of Ynare (using part(a))
f(x) =
91 (9₁) = 2ny₁ (1 - y²)¹-¹, 0<y₁ <1, In (Yn) = 2ny2n-1, 0<yn <1.
Q4(c) The joint probability density function of Y₁ < Y₂ < Y3 <... Yn is given by
0 < x < 1,
otherwise.
g(y₁, Y2,..., Yn) = n!f(y₁)ƒ(Y2) ··· ƒ(Yn), Y₁ <Y2 <··· < Yn-
Suppose n = 3, write down the joint density g(y₁, 92, 93) of Y₁ < Y₂ < Y3 when
0 < x < 1,
otherwise.
91,n (y1, yn) =
f(x) =
2x,
0
. Now derive the marginal probability density of g₁ (3₁) of Y₁ and 93 (43) of Y3 and verify that it matches with the answer in part(b) for n = 3.
Q4(d) Note that the joint probability density function of (Y₁, Yn) is given for the special case of f(x) by
n!
¡ (2y₁)[y? — y?]”¯²(2yn), 0<Y₁ <Yn <1.
(n − 2)!
Write down the joint density function of 9₁,3(y₁, y3) of Y₁, Y3 when n = 3. Let Z₁ = Y - Y₁ and Z2 = Y3. Find the joint density of
(Z₁, Z₂) using the transformation method and deduce the probability density function of Z₁ which is the range of the random sample X₁,
X₂, and X3.
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