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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

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
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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.
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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