A child gets a weekly allowance (a certain non-negative number of dollars). The amount (in dollars) is random with expectation equal to 6 and variance equal to 6. Let’s denote this random variable by X. I want to estimate (from above) the chances that the child gets 10 dollars or more on a given week. In other words, I want to find c such that P(X ≥ 10) ≤ c, with c that is as small as possible. (a) Use the Markov inequality to find c such that P(X ≥ 10) ≤ c. (b) Use the Chebyshev inequality for the variance to find c such that P(X ≥ 10) ≤ c. (c) Read about the one-sided Chebyshev inequality, and use it to find c such that P(X ≥ 10) ≤ c.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
A child gets a weekly allowance (a certain non-negative number of dollars). The amount (in dollars) is random with expectation equal to 6 and variance equal to 6. Let’s denote this random variable by X. I want to estimate (from above) the chances that the child gets 10 dollars or more on a given week. In other words, I want to find c such that P(X ≥ 10) ≤ c, with c that is as small as possible.
(a) Use the Markov inequality to find c such that P(X ≥ 10) ≤ c.
(b) Use the Chebyshev inequality for the variance to find c such that P(X ≥ 10) ≤ c.
(c) Read about the one-sided Chebyshev inequality, and use it to find c such that P(X ≥ 10) ≤ c.
Given that
The expectation = 6
The variance = 6
The child wants to find P(X using
- the Markov inequality
- The Chebyshev inequality for the variance
- The one-sided Chebyshev inequality
Markov inequality:
If X is a random variable then the Markov inequality for discrete and mixed random variables
P(X
Chebyshev's inequality:
If X is a random variable and a>0 then Chebyshev's inequality is
One-Sided Chebyshev: For any positive number a > 0, the following one-sided Chebyshev inequalities hold
P(X
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