![9. (10 points) Recall the random variable X in question 4. Sx = [-6,3] and f(t) = x2/81 for reSx (a) (5 points) Using Chebysh](http://img.homeworklib.com/questions/7a431410-e2f8-11ea-9704-f13d1564a7e8.png?x-oss-process=image/resize,w_560)
(b) (5 points) Calculate this probability exactly and compare to the bound found in part (a). Is this a helpful bound in this case?
Homework Answers
![Given, X is randan variable, with pdf . fenomen - 64x4 3 a Using Chekeyshews Inequality P[x² + 1st +14 > 3.9375] P[ x?+3.45](http://img.homeworklib.com/questions/7af2ece0-e2f8-11ea-b46b-615eb0ea6890.png?x-oss-process=image/resize,w_560)
![New. € (x + 7 = t[x2] + (+) 2 + 15 EC) - 3 x² du 21. 6/ -6 81 1296 ( 43 -3.75 81x4 81X 4 3 xe du E[x2] 81 3 19.8 [x57 -6 81x5](http://img.homeworklib.com/questions/7c5be050-e2f8-11ea-b913-f7425d9e61f0.png?x-oss-process=image/resize,w_560)
![Exact probability x2 & 15x + 14 > 3.9375 25] 2 =P IS =P [X + << -2] + P[*++> >>] = PL < -5.75) + P[ * > -1.75) =I- P[-5.75<x](http://img.homeworklib.com/questions/7d916960-e2f8-11ea-a305-4fb3ef133f38.png?x-oss-process=image/resize,w_560)
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(b) (5 points) Calculate this probability exactly and compare to
the bound found in part (a)....
5. Let X > 0 be a random variable with EX = 10 and EX2 =...
5. Let X > 0 be a random variable with EX = 10 and EX2 = 140. a. Find an upper bound on P(X > 14) involving EX using Markov's inequality. b. Modify the proof of Markov's inequality to find an upper bound on P(X > 14) in- volving EX? c. Compare the results in (a) and (b) above to what you find from Chebyshev's inequality.
Exercise 2 Consider a random variable X with E]5 and VarX 16 (a) Calculate P(lz-5 <...
Exercise 2 Consider a random variable X with E]5 and VarX 16 (a) Calculate P(lz-5 < 6) if X follows a normal distribution. (b) Use Chebyshev's inequality to provide a lower bound for P(-5). (No longer assume X is normal.)
(b) (5 points) Using CLT, approximate the probability that P(X = 18). (c) (5 points) Calculate...
(b) (5 points) Using CLT, approximate the probability that P(X =
18).
(c) (5 points) Calculate P(X = 18) exactly and compare to
part(b).
8. (15 points) Let X~Binomial(30,0.6). (a) (5 points) Using the Central Limit Theorem (CLT), approximate the probability that P(X > 20).
Please show your work with a brief but logical explanation. Suppose X is a random variable with p(X 0) 4/5, p(X-1) 1/10, p(X-9) 1/10. Then (a) Compute Var [X] and B [X] (b) What is the upper bound on...
Please show your work with a brief but logical explanation.
Suppose X is a random variable with p(X 0) 4/5, p(X-1) 1/10, p(X-9) 1/10. Then (a) Compute Var [X] and B [X] (b) What is the upper bound on the probability that X is at least 20 obained by applying Markov's inequality? c) What is the upper bound on the probability that X is at least 20 obained by applying Chebychev's inequality'?
Suppose X is a random variable with p(X...
This is Probability and Statistics in Engineering and Science Please show your work! especially for part B A Poisson distribution with λ=2 X~Pois(2) A binomial distribution with n=10 and π=0.45. X~bin...
This is Probability and Statistics in Engineering and
Science
Please show your work! especially for part B
A Poisson distribution with λ=2 X~Pois(2)
A binomial distribution with n=10 and π=0.45.
X~binom(10,0.45)
Question 4. An inequality developed by Russian mathematician Chebyshev gives the minimum percentage of values in ANY sample that can be found within some number (k21) standard deviations from the mean. Let P be the percentage of values within k standard deviations of the mean value. Chebyshev's inequality states...
Problem 2. Suppose the sample space S consists of the four points and the associated probabilities...
Problem 2. Suppose the sample space S consists of the four points and the associated probabilities over the events are given by P(cu 1)-0.2, P(ω2)-0.3, P(ag)-0.1, P(04)-0.4 Define the random variable X1 by and the random variable X2 by X2(2) 5, (a) Find the probability distribution of X1 (b) Find the probability distribution of the random variable X1 +X2 Problem 3. The random variable X has density function f given by 0, elsewhere (a) Assuming that 0.8, determine K (b)...
Probability & Statistics (25 points) 1. (5 points) If the probability that student A will fail...
Probability & Statistics (25 points) 1. (5 points) If the probability that student A will fail a certain statistics examination is 0.3, the probability that student B will fail the examination is 0.2, and the probability that bosh student A and student B will fail the examination is 0.1. a) What is the probability that at least one of these two students will fail the examination? b) What is the probability that exactly one of the two students will fail...
Part III – Probability and Statistics Each question is worth 4 points. 1. Consider the following...
Part III – Probability and Statistics Each question is worth 4 points. 1. Consider the following experiment and events: two fair coins are tossed, E is the event "the coins match”, and F is the event “at least one coin is Heads”. (a) Find the probabilities P(E), P(F), P(EUF), and P(En F). (b) Are the events and F independent? Explain. 2. Let X be a discrete random variable with the probability function given by f(2) k(x2 – 2x) + 0.2...
(i) Show that 15 (ii) Show that (X) 5/12 and E(Y) 5/8 3(1 - 2X2 +X4)...
(i) Show that 15 (ii) Show that (X) 5/12 and E(Y) 5/8 3(1 - 2X2 +X4) 4(2- 3X +X3) (iii) Show that 3(y|X) (iv) Verify thatE(Y)E(Y) 14] 7. (a) State Chebyshev's inequality and prove it using Markov's inequality 15] (b) Let (2, P) be a probability space representing a random experiment that can be repeated many times under the same conditions, and let A C S2 be a random event. Suppose the experiment is repeated n times (i) Write down...
LSM 5 Part C. Suppose that you are walking on a straight line. You start at position X, -0, and only walk in the positive direction. Your positions afcer taking the ith step is denoted by X,. For...
LSM 5 Part C. Suppose that you are walking on a straight line. You start at position X, -0, and only walk in the positive direction. Your positions afcer taking the ith step is denoted by X,. For each step, your step size, denoted by S, = X,-X,-ı , is a random variable uniformly distributed between 1 foot and 2 eet. Assume that sizes of different steps are mutually independent. 8. (10 credits) Let Xy be your position after taking...
5. Let X > 0 be a random variable with EX = 10 and EX2 = 140. a. Find an upper bound on P(X > 14) involving EX using Markov's inequality. b. Modify the proof of Markov's inequality to find an upper bound on P(X > 14) in- volving EX? c. Compare the results in (a) and (b) above to what you find from Chebyshev's inequality.
Exercise 2 Consider a random variable X with E]5 and VarX 16 (a) Calculate P(lz-5 < 6) if X follows a normal distribution. (b) Use Chebyshev's inequality to provide a lower bound for P(-5). (No longer assume X is normal.)
(b) (5 points) Using CLT, approximate the probability that P(X =
18).
(c) (5 points) Calculate P(X = 18) exactly and compare to
part(b).
8. (15 points) Let X~Binomial(30,0.6). (a) (5 points) Using the Central Limit Theorem (CLT), approximate the probability that P(X > 20).
Please show your work with a brief but logical explanation.
Suppose X is a random variable with p(X 0) 4/5, p(X-1) 1/10, p(X-9) 1/10. Then (a) Compute Var [X] and B [X] (b) What is the upper bound on the probability that X is at least 20 obained by applying Markov's inequality? c) What is the upper bound on the probability that X is at least 20 obained by applying Chebychev's inequality'?
Suppose X is a random variable with p(X...
This is Probability and Statistics in Engineering and
Science
Please show your work! especially for part B
A Poisson distribution with λ=2 X~Pois(2)
A binomial distribution with n=10 and π=0.45.
X~binom(10,0.45)
Question 4. An inequality developed by Russian mathematician Chebyshev gives the minimum percentage of values in ANY sample that can be found within some number (k21) standard deviations from the mean. Let P be the percentage of values within k standard deviations of the mean value. Chebyshev's inequality states...
Problem 2. Suppose the sample space S consists of the four points and the associated probabilities over the events are given by P(cu 1)-0.2, P(ω2)-0.3, P(ag)-0.1, P(04)-0.4 Define the random variable X1 by and the random variable X2 by X2(2) 5, (a) Find the probability distribution of X1 (b) Find the probability distribution of the random variable X1 +X2 Problem 3. The random variable X has density function f given by 0, elsewhere (a) Assuming that 0.8, determine K (b)...
Probability & Statistics (25 points) 1. (5 points) If the probability that student A will fail a certain statistics examination is 0.3, the probability that student B will fail the examination is 0.2, and the probability that bosh student A and student B will fail the examination is 0.1. a) What is the probability that at least one of these two students will fail the examination? b) What is the probability that exactly one of the two students will fail...
Part III – Probability and Statistics Each question is worth 4 points. 1. Consider the following experiment and events: two fair coins are tossed, E is the event "the coins match”, and F is the event “at least one coin is Heads”. (a) Find the probabilities P(E), P(F), P(EUF), and P(En F). (b) Are the events and F independent? Explain. 2. Let X be a discrete random variable with the probability function given by f(2) k(x2 – 2x) + 0.2...
(i) Show that 15 (ii) Show that (X) 5/12 and E(Y) 5/8 3(1 - 2X2 +X4) 4(2- 3X +X3) (iii) Show that 3(y|X) (iv) Verify thatE(Y)E(Y) 14] 7. (a) State Chebyshev's inequality and prove it using Markov's inequality 15] (b) Let (2, P) be a probability space representing a random experiment that can be repeated many times under the same conditions, and let A C S2 be a random event. Suppose the experiment is repeated n times (i) Write down...
LSM 5 Part C. Suppose that you are walking on a straight line. You start at position X, -0, and only walk in the positive direction. Your positions afcer taking the ith step is denoted by X,. For each step, your step size, denoted by S, = X,-X,-ı , is a random variable uniformly distributed between 1 foot and 2 eet. Assume that sizes of different steps are mutually independent. 8. (10 credits) Let Xy be your position after taking...
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