Ask Question
30 October, 22:58

A lake contains 4 distinct types of fish. Suppose that each fish caught is equally likely to be any one of these types. Let Y denote the number of fish that need be caught to obtain at least one of each type. a. Give an iterval (a, b) such that P{a≤Y≤b} ≥ 0.90b. Using the one-sided Chebyshev inequality, how many fish need we plan on catching so as to be at least 90 percent certain of obtaining at least one of each type.

+2
Answers (1)
  1. 30 October, 23:06
    0
    a) P (μ-k*σ≤ Y ≤ μ+k*σ) ≥ 0.90

    a = μ-3.16*σ, b = μ+3.16*σ

    b) P (Y≥ μ+3*σ) ≥ 0.90

    b = μ+3*σ

    Step-by-step explanation:

    from Chebyshev's inequality for Y

    P (| Y - μ|≤ k*σ) ≥ 1-1/k²

    where

    Y = the number of fish that need be caught to obtain at least one of each type

    μ = expected value of Y

    σ = standard deviation of Y

    P (| Y - μ|≤ k*σ) = probability that Y is within k standard deviations from the mean

    k = parameter

    thus for

    P (| Y - μ|≤ k*σ) ≥ 1-1/k²

    P{a≤Y≤b} ≥ 0.90 → 1-1/k² = 0.90 → k = 3.16

    then

    P (μ-k*σ≤ Y ≤ μ+k*σ) ≥ 0.90

    using one-sided Chebyshev inequality (Cantelli's inequality)

    P (Y - μ≥ λ) ≥ 1 - σ² / (σ²+λ²)

    P{Y≥b} ≥ 0.90 → 1 - σ² / (σ²+λ²) = 1 - 1 / (1 + (λ/σ) ²) = 0.90 → 3 = λ/σ → λ = 3*σ

    then for

    P (Y≥ μ+3*σ) ≥ 0.90
Know the Answer?
Not Sure About the Answer?
Find an answer to your question 👍 “A lake contains 4 distinct types of fish. Suppose that each fish caught is equally likely to be any one of these types. Let Y denote the ...” in 📗 Mathematics if the answers seem to be not correct or there’s no answer. Try a smart search to find answers to similar questions.
Search for Other Answers