1) Consider a random sample X₁, X₂,...,X, from a population distributed with the following mass function: f(x) = 3% e x! for x = 0,1,....,00 elsewhere d) What is the probability that Y = EX₁ is at least 3?
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- Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2If X and Y are random variables and X is a geometric random variable where p = 0.1 then what is the probability mass function of Y = sin(X*pi) ?
- Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 01) Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the estimator of moments for the parameter θ.Consider a random variable X with E[X] = 10, and X being positive. Estimate E[ln√X] using Jensen’s inequality.
- Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the temperature (◦F) at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint densitySuppose that X1, X2, X3 are independent with the common probability mass function: P{Xi = 0} = 0.2, P{Xi =1} = 0.3, P{Xi = 3} = 0.5 i =1, 2,3 a. Plot the probability mass function of X2_average = (X1 + X2)/ 2 b. Determine E [X2_average] and Var [X2_average] c. Plot the probability mass function of X3_average = (X1 + X2 + X3)/ 3 d. Determine E [X3_average] and Var [X3_average]The number of trams X arriving at the St. Peter's Square tram stop every t minutes has the following probability mass function: p(x) =(0.25t)^x/x! * exp(-0.25t) for x=0,1,2,... the probability that 3 to 5 trams arrive in a 6 minute period is