Suppose you have the following random sample with n independent obser- vations: iid X1, X2,..., Xn Uniform(-0,0) (a) the minimum of the sample, denoted by X(1)· Find the cumulative distribution function and probability density function of (b) What does X(1) converge in probability to? Justify your answer. What does E=1 Xi (c) converge in probability to? Justify your answer. nX(1)
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- 1) Suppose that Y₁, Y2,...,Y₁ is a random sample from a population with probability density function 1,7, f(x)=B³ Find the Maximum Likelihood Estimator of B. 00 elsewhere1. Determine the constant c so that the function p(x) = 0(x) defines the probability den- sity for a random variable {. Then find the cumulative distribution function and the mean of {. %3DSuppose that the unknown X is uniformly distributed between 0 and 2. What is the expected value of (4X3 + 1)?
- Q1. The cumulative distribution function of the random variable X is F(x) = 0, (x+1)², 1- (1-2)², 1, for x 1 (a) Compute-P(0.5 < X <1)¶ (b) What is the probability density function of X.?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.A product is classified according to the number of defects, it contains (X1) and the factory that produces it (X2),. The joint probability distribution is given by X/X2 2 0 1/8 1/16 I/16 1/16 3/16 1/8 3 1/8 1/4 (a) Find the marginal distribution of X1
- Previously, De Anza's statistics students estimated that the amount of change daytime statistics students carry is exponentially distributed with a mean of $0.88. Suppose that we randomly pick 25 daytime statistics students. Find the probability that an individual had between $0.80 and $1.00. Graph the situation, and shade in the area to be determined.Exercise 24 Let A and B be 2 events : P(A) : 1 and P(B') 1 Can A and B be mutually exclusive? Why?The differentiation approach to derive the maximum likelihood estimator (mle) is not appropriate in all the cases. Let X₁, X2,,X₁ be a random sample of size n from the population of X. Consider the probability function of X fe-(2-0), if 0The maintenance department in a factory claims that the number of breakdowns of a particular machine follows a Poisson distribution with a mean of 2 breakdowns every 428 hours. Let x denote the time (in hours) between successive breakdowns. (a) Find λ and Ux. (Write the fraction in reduced form.) ux = f(x) = 214 (b) Write the formula for the exponential probability curve of x. P(x <4) ✔ Answer is complete and correct. 1 P(115Suppose X is a discrete random variable which only takes on positive integer values. For the cumulative distribution function associated to X the following values are known: F(23) 0.34 F(29) = =0.38 F(34) 0.42 F(39) 0.47 F(44) = 0.52 F(49) 0.55 F(56) = 0.61 = Determine Pr[29Let Y1, Y2,., Ya be a collection of independent random variables with distribution function y 8 Show that Y converges in probability to a constant, and provide that constant. 1SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON