Chebyshev\'s Inequality. This inequality points out another useful property of t
ID: 3329286 • Letter: C
Question
Chebyshev's Inequality. This inequality points out another useful property of the standard deviation In particular, it states that "That the probability that any random variable X falls within k standard deviations of its means is at least 1 - 1/k2", i.e For example,if we know that X has mean 3 and standard deviation 1, then we can conclude that the probability that X lies between 1 and 5 is at least 1 1/22 0.75 1. Let X denote the amount of rainfall received per week in a region. Assume that the mean of rainfall per week is 1.00 inch and the variance is 0.0625 inch. What distribution is assumed? Would it be unusual for this region to receive more than 2 inches of rain in a given week? Explain this on the basis of Chebyshev's inequality 2. Let X be the number of cases of rabies reported in a given state per week. Assume that the average of number of cases of rabies is1 and the variance is 1/25. What distribution is assumed? Would it be unusual to observe two cases in a given week? Explain this using the basis of Chebyshev' inequality.Explanation / Answer
(1) Mean rainfall per week = 1.00 inch
Varaince per week = 0.0625 inch2
Standard Deviation per week = 0.25 inch
A normal distribution can be assumed.
Pr(Weekly Rainfall > 2 inch) = ?
as per Chebyshev's inequaity
P( l X - l <= k) <= (1 - 1/k2 )
here k = 1/0.25 = 4
P(0 inch < Rainfall < 2 inch) < = (1 - 1/42)
P(0 inch < Rainfall < 2 inch) <= 0.9375
P(Rainfall > 2 inch) = 1 - 0.9375 = 0.0625
so, this is not quite unusual to have 2 inches of rain in a given week as it has at least 6.25% probability to have more than 2 inch of rain.
(2) AVerage number of cases of rabies = 1
Variance = 1/25
standard deviation = 1/5 = 0.2
The distribution assumed is a normal approximation to binomial.
as per Chebyshev's inequaity
P( l X - l <= k) <= (1 - 1/k2 )
here k = 1/0.2 = 5
P(0 < rabies case < 2 ) < = (1 - 1/52)
P(0 < Rabies case < 2 inch) <= 0.96
P(Rabies case > 2 ) = 1 - 0.96 = 0.04
so, this is quite unusual to have more than 2 cases of rabies per week. As it has at max 4% probability to that to occur that is lesss than 0.05 so it is unusual to observe two cases in a given week.