1. Suppose we have a family of distribution indexed by a parameter k where k is any positive integer. Let F₁(x) = xk be the c.d.f. where 0 ≤ x ≤ 1 (a) Find the corresponding p.d.f. or p.m.f., whichever is appropriate. (b) Find the mean of the distribution with c.d.f. Fk. (c) Find the variance of the distribution with c.d.f. Fk. (d) Find the SCV.
1. Suppose we have a family of distribution indexed by a parameter k where k is any positive integer. Let F₁(x) = xk be the c.d.f. where 0 ≤ x ≤ 1 (a) Find the corresponding p.d.f. or p.m.f., whichever is appropriate. (b) Find the mean of the distribution with c.d.f. Fk. (c) Find the variance of the distribution with c.d.f. Fk. (d) Find the SCV.
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Transcribed Image Text:1. Suppose we have a family of distributions indexed by a parameter \( k \) where \( k \) is any positive integer. Let \( F_k(x) = x^k \) be the c.d.f. where \( 0 \leq x \leq 1 \).
(a) Find the corresponding p.d.f. or p.m.f., whichever is appropriate.
(b) Find the mean of the distribution with c.d.f. \( F_k \).
(c) Find the variance of the distribution with c.d.f. \( F_k \).
(d) Find the SCV.
Expert Solution
Step 1
Fk(x) = xk ; 0≤x≤1
The CDF of X is given by,
Note that, it is a continuous random variable. So, It have a Probability Density Function (PDF) not Probability Mass Function (PMF).
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