2 of 3 01- 5. Suppose X is a discrete random variable that has a geometric...
2 5. Suppose X is a discrete random variable that has a geometric distribution with p= a. Compute P(X > 6). [5] b. Use Markov's Inequality to estimate P(X > 6). [5] c. Use Chebyshev's Inequality to estimate P(X > 6). [5]
5. Suppose X is a discrete random variable that has a geometric distribution with p= 1. a. Compute P(X > 6). [5] b. Use Markov's Inequality to estimate P(X> 6). [5] c. Use Chebyshev's Inequality to estimate P(X>6). [5] t> 0 6. Let be the probability density function of the continuous 0 t< 0 random variable X. a. Verify that g(t) is indeed a probability density function. [8] b. Find the median of X, i.e. the number m such that...
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.
Let X be a discrete random variable that follows a Poisson distribution with = 5. What is P(X< 4X > 2) ? Round your answer to at least 3 decimal places. Number
[Q#2] (7pts) Suppose a discrete random variable Y has a Geometric probability distribution with probability of success p (>0). Its p.d.f. p(y) is defined as P(Y = y) = p(y) = p (1-p)y-1 for y = 1,2,3, ... Verify that the sum of probabilities when the values of random variable Y are even integers only is 1-p. That is to find p(2) + p(4) +p(6) +.. 2 – p
Problem 8 (4x4 pts) Suppose Xi, X2-, ..,. Xn are each independent Poisson random variables with mean 1. Let 100 k=1 (a) Rccall, Markov's incquality is P(Y > a) for a> 0 Using Markov's inequality, estimate the probability that P(Y > 120). (b) Rccal, Chebyshev's incquality is Using Chebyshev's inequality, estimate P( Y-?> 20) (c), (d) Using the Central Limit Theorem, estimate P(Y > 120) and Ply-? > 20).
A random variable, x, has a normal distribution with u = 11.6 and 6 = 2.50. Determine a value, Xo, so that a. P(x>xo) = 0.05. b. P(xsxo) = 0.975. c. P(H-Xo xxx H +0) = 0.95. a. Xo = (Round to one decimal place as needed.) b. Xo (Round to one decimal place as needed.) C. XO (Round to one decimal place as needed.)
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.)
Let X be a random variable following a continuous uniform distribution from 0 to 10. Find the conditional probability P(X >3 X < 5.5). Chebyshev's theorem states that the probability that a random variable X has a value at most 3 standard deviations away from the mean is at least 8/9. Given that the probability distribution of X is normally distributed with mean ji and variance o”, find the exact value of P(u – 30 < X < u +30).
2. A random variable X has a cdf given by F(x) = 1 . x < 0 0 < x < 1 <3 x > 3 11, (f) What is P(X = 1)? (g) Find E(X), the expectation of X. (h) Find the 75th percentile of the distribution. Namely, find the value of 70.75 SO that P(X < 70.75) = F(710.75) = 0.75. (i) Find the conditional probability P(X > X|X > 3).