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This exercise demonstrates that, in general, the results provided by Tchebysheff's theorem cannot be improved upon. Let Ybe a random variable such that p(-1) = . p(1) = . p (0) = %3D a Show that E(Y) = 0 and V(Y) = 1/9. b Use the probability distribution of Yto calculate P (|Y – µ| 30). 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, and 1, with p(-1)% 3 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 (X- ux|2 2ox) = 1/4. f any k>1 is specified, how can a random variable Wbe constructed so that P (W-HwlZ kow) = 1/k2?

This exercise demonstrates that, in general, the results provided by Tchebysheff's theorem cannot be improved upon. Let Ybe a random variable such that p(-1) = . p(1) = . p (0) = %3D a Show that E(Y) = 0 and V(Y) = 1/9. b Use the probability distribution of Yto calculate P (|Y – µ| 30). 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, and 1, with p(-1)% 3 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 (X- ux|2 2ox) = 1/4. f any k>1 is specified, how can a random variable Wbe constructed so that P (W-HwlZ kow) = 1/k2?

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 31CR
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This exercise demonstrates that, in general, the results provided by Tchebysheff's theorem cannot be improved upon. Let Ybe a random variable such that p(-1) = –, p (0) = º, p(1) == 18 P(1) = 18 a Show that E(Y) = 0 and V(Y) = 1/9. b Use the probability distribution of Yto calculate P (|Y – el2 30). 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, and 1, 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 (X- ux| 2ox) = 1/4. *d If any k> 1 is specified, how can a random variable Wbe constructed so that P (W - wl2 kow) = 1/k-?
Transcribed Image Text:This exercise demonstrates that, in general, the results provided by Tchebysheff's theorem cannot be improved upon. Let Ybe a random variable such that p(-1) = –, p (0) = º, p(1) == 18 P(1) = 18 a Show that E(Y) = 0 and V(Y) = 1/9. b Use the probability distribution of Yto calculate P (|Y – el2 30). 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, and 1, 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 (X- ux| 2ox) = 1/4. *d If any k> 1 is specified, how can a random variable Wbe constructed so that P (W - wl2 kow) = 1/k-?
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