1. Consider a random variable X with probability density function given as: 0 if x < 0, 1 if 0 ≤ x ≤ 1, f(x) 2 1 if 1 < <∞. -2x² Verify if f(x) is indeed a density function. If it is, find the distribution function of X.
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- b) Let X₁, X2, X3,...,Xn be a random sample of n from population X distributed with the following probability density function: f(x;0)=√√2n0 0, -20₁ if -∞0 < x <∞0 otherwise (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. (iii) Check whether or not the estimator obtained in (ii) is unbiased. (iv) Find the Fisher information in this sample of size n about the parameter 0.Suppose that X is a continuous random variable with a probability density function is given by f(x)= 25 when x is between -2 and 2, and f(x)=0 otherwise. a.)Find E(X2), where X is raised to the power 2 b.) Find Var(2X+2)Let XE {-1,0, 1} . That is, X is a discrete random variable only takes three values -1, 0, and 1. Suppose the equality for Chebyshev's inequality holds for X and P(X = 0) = 0.3 , find P(X = 1) Let X be a random variable with probability density f(x) = x6 for x > 1 and O else. Use Chebyshev's inequality to bound P(X > 2.5) . Round your answer to 3 decimal places.
- The probability density function for the continuous random variable X is given by: (А(х? — 2х + 21) 0(Continuous random variable) In a city, the daily electricity consumption (millions of kWh) is a random variable whose probability density is: f(x) - ਵਿਆ = 0 x xe 3 six>0 six<0 The city's electric power plant has a daily capacity of 12 million kWh. a. Verify that the given function is a probability density function. b. Calculate the probability that the power plant cannot meet the demand for electric power.The probability that none of the three bulbs in a set of traffic lights will have to be replaced during the first 1200 hours of operation if the lifetime X of a bulb (in thousands of hours) is a random variable with probability density function f(x) = 6[0.25-(x-1.5)2] when 1 ≤ x ≤ 2 and f(x) = 0 otherwise isRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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