Let Y be a continuous random variable with a probability mass function: - = r· | Derive the moment generating function (MGF) of a random variable Y.
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- Find the moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 < x < 1 0 elsewhere and use it to find μ’1,μ’2, and σ^2.Let x be a continuous random variable with the density function: f(x) = 3e-3x when x>0 and 0 else Find the variance of the random variable x.Suppose the random variables X and Y have joint probability density function f(x,y) given by: (image)Find: P(X < Y) = fX|Y=y (x)
- Suppose that X, Y are jointly continuous with joint probability density function f( x, y){ xe^-x(1+y), ifx >0 and y >00, otherwise. (a) Find the marginal density functions of X and Y. (b) Calculate the expectation E[XY]. (c) Calculate the expectation EIX/(1+ Y )1. (e) Determine if the random variables X and Y in this exercise are independent.Suppose that the continuous random variable X has CDF Fx(X) = {(x-2)/x , x>2 and 0, x=<2} a. Determine, and sketch, the pdf (probability density function) of X. b. Find the mean and variance of X c. Determine the pdf of the random variable Y=X^2A continuous random Variable X has probability Density function defined by f(x) = 5-5x; 0
- On a production line, parts are produced with a certain average size, but the exact size of each part varies due to the imprecision of the production process. Suppose that the difference between the size of the pieces produced (in millimeters) and the average size, which we will call production error, can be modeled as a continuous random variable X with a probability density function given by f(x) = 2, 5e^(-5|x|), for x E R (is in the image). Parts where the production error is less than -0.46 mm or greater than 0.46 mm should be discarded. Calculate (approximating to 4 decimal places): a) What is the proportion of parts that the company discards in its production process? b) What is the proportion of parts produced where the production error is positive? c) Knowing that for a given part the production error is positive, what is the probability of this part being discarded?Let X and Y be two continuous random variables with joint probability density function f(x,y) = 2xy for 0 < x < y < 1. Find the covariance between X and Y.Suppose a continuous random variable X~Fx(x): f(x,y) = {1/4e^-1x/4, if x≥0 0, x<0} What is the cumulative density function of Y=min{2,X}?
- X is a random variable whose cumulative distribution function F(x) satisfies F(1) = 13, F(2) = 34, and F(3) = 1 • Make X discrete and give its probability mass function.• Make X continuous and give its probability density function.Let X be a discrete random variable with probability mass function P(X= x) =p(1 −p)^x ; x= 0,1,2,.... Here p∈[0,1]. Calculate the moment generating function (MGF) of X, the mean, and variance of this distribution (using the MGF).The probability density function of the random variable X is as in the picture with λ> 0. Find the moments estimator (λ^) of the parameter λ.