Answered: Chebyshev’s theorem provides that for a…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Related questions

Question

Chebyshev’s theorem provides that for a random variable X with any
probability distribution and a positive constant k greater than or equal
to 1, the inequality P (|X − μ| ≥ kσ) ≤ 1
k2 holds. This signifies that
the probability of X deviating from its mean by at least k standard
deviations is at most 1
k2 .
a. Determine the upper limit for k being 2, 3, 4, 5, and 10.
b. Calculate the mean μ and standard deviation σ for the following
distribution. Then, assess P (|X − μ| ≥ kσ) for the given k values
and compare these to the upper bounds found previously.
x 0 1 2 3 4 5 6
p(x) .1 .15 .2 .25 .06 .04
2

c. Assume X can take on the values -1, 0, and 1 with probabilities
1
18 , 8
9 , and 1
18 respectively. Estimate P (|X − μ| ≥ 3σ), and relate
it to the corresponding upper limit.
d. Propose a probability distribution where P (|X − μ| ≥ 5σ) is 0.04.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Expert Solution