Question
For the distribution of Exercise 1, calculate $P(|X-\mu| \geq 2)$ and compare this with the upper bound for this probability obtained from Tchebycheff's inequality.
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We need to know the mean (\(\mu\)) and the variance (\(\sigma^2\)) of the random variable \(X\) in order to proceed with the calculations. Show more…
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This exercise demonstrates that, in general, the results provided by Tchebysheff's theorem cannot be improved upon. Let $Y$ be a random variable such that $$p(-1)=\frac{1}{18}, \quad p(0)=\frac{16}{18}, \quad p(1)=\frac{1}{18}$$ a. Show that $E(Y)=0$ and $V(Y)=1 / 9$ b. Use the probability distribution of $Y$ to calculate $P(|Y-\mu| \geq 3 \sigma) .$ Compare this exact probability with the upper bound provided by Tchebysheff's theorem to see that the bound provided by Tchebysheff's theorem is actually attained when $k=3$ *.c. In part.(b) we guaranteed $E(Y)=0$ by placing all probability mass on the values $-1,0,$ and1, with $p(-1)=p(1) .$ The variance was controlled by the probabilities assigned to $p(-1)$ and $p(1) .$ Using this same basic idea, construct a probability distribution for a random variable $X$ that will yield $P\left(\left|X-\mu_{X}\right| \geq 2 \sigma_{X}\right)=1 / 4$ * d. If any $k>1$ is specified, how can a random variable $W$ be constructed so that $P\left(\left|W-\mu_{W}\right| \geq k \sigma_{W}\right)=1 / k^{2} ?$
Discrete Random Variables and Their Probability Distributions
Tehebysheff’s Theorem
Suppose X1, ..., X10 are independent samples from distribution F with expectation 5 and variance 3. Use the Central Limit Theorem to estimate the probability that X bar is outside the interval 4.0 and 5. Compare to the bound obtained using Chebyshev's inequality.
(1) (Similar to textbook 13.1) In this problem, you'll explore Chebyshev's inequality for particular distributions. Fill in the following table of values of P(|X - μ| < kσ) for various values of k and various distributions. Start by calculating the mean and standard deviation for each distribution. The bounds (from Chebyshev's inequality) are at the top; make sure all your answers are consistent with the theoretical bound! Distribution | k = 2 | k = 3 | k = 4 U(-1, 1) Par(3) N(0, 1) Bin(10, 0.5) Bin(20, 0.7) Chebyshev bound: | 75% | 88.8% | 93.75% Note that quick exercise 13.2 is very similar to this but for Exp(1).
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