- Question 7 For Some T 1 Let X Be A Random Variable Taking The Values 0 And T With Probabilities 1 1 P X 0 1 P 1 (40.92 KiB) Viewed 27 times
Question 8. ♣ We still keep the notations of Question 7. For every a in R, define the probability I(t) := P(|X − 1| > a). Chebyshev's inequality provides an upper bound of It). Besides, when one can com- pute exactly It), we can compute the difference between the upper bound obtained by Chebyshev's inequality and I). We call it the Chebyshev gap for X, at a and for t. We denote it e(t) (a). = Hence, it is easy to see that, if t = 10 and a = 8, then (¹0) 1/10 and that Chebyshev's inequality provides 9/64, as an upper bound of I(10). Hence, the Chebyshev gap for X, at a = 8 and for t = 10 is equal to 9/64 - 1/10 = 9/80. In other words, one can write: (10) 9 80 ≈ 0.04. □ C ☐ (¹0) = 7/12. ☐ (¹0) = 1/25. 10) 3 □ C = □ (¹0) (5) = ☐ (¹0) = 12/7. 10 10 None of these answers are correct ₂(10) 13 50 Question 9. We still keep the notations of Question 7 to 8. One can affirm that: (10) None of these answers are correct ☐ = 0. □ (¹0) = 2/9. 7 ☐ (10) = 1/10. ¹10 (10) 10* 9 3 2 □ (10(10): = □ (10) 100 100 25 Question 10. We still keep the notations of Question 7 to 9. More generally, for any t in (1, ∞), one can affirm that: 1 □ I(t) = 0, Va> t. ☐ I(t) = 0, Va<t. □ I(t) = Va> t > 1. = Va > t > 1. t2³ □ (10) = 1/3. □ (¹0) (10) = None of these answers are correct ☐ ¹0) = 1/10. (10) (10)
Question 11. We still keep the notations of Question 7 to 9. More generally, for any t in (1, ∞), one can affirm using Chebychev's inequality, that: ■ [(¹) < 22, Va>t≥1. □ I(¹) < 7, Vt > a. ☐ I(t) 1, Va>t. a t None of these answers are correct ☐ (¹) < Va> t. a "One might have to use the inequality t − 1 ≤ a, valid for all a > t. Question 12. We still keep the notations of Question 7 to 11. One wonders here if one can find a gap smaller than a certain number. One can affirm that: None of these answers are correct □ One can find a gap smaller than 10-2. □ One can find a gap smaller than 10-P, for all p in N*. One can find a gap smaller than 10-³. ☐ There are some p in N*, for which one can not find a gap smaller than 10¯º, for all p in N* One can not find a gap smaller than 10-7. Question 13. (Optional) We still keep the notations of Question 7 to 12. One can affirm that: None of these answers are correct ☐ One can not improve Chebychev Inequality, in general. □ One can improve Chebychev Inequality, in general.