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- A continuous random variable X has a pdf of the form: f(x) = (824/9) x^2, for 0.01 < X < 0.32. Calculate the standard deviation (sigma) of X.What is the Marginal PDF of f(x)=2? Is the event independent?1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.
- Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?X follows a gamma distribution with PDF f(x) = 4xe-2x , where X > 0(a) Derive E(Xn ).If Y is a continuous, uniformly distributed random variable over the interval(4,10), then the value of the PDF between 4 and 10 is?
- Consider a random variable X with E[X] = 10, and X being positive. Estimate E[ln√X] using Jensen’s inequality.suppose x has an exponential distribution with probability density function f(x) =2e^-2x, x>0. Then P(X>1)1)Let x be a random variable Gaussian with zero mean and variance 1. Find:a)The conditional pdf and pdf of x given x > 0;b)E [ x| x>0 ]c)Var [ x | x >0]
- 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 f(x) = ½ , -1 < x < 1 0 otherwise be a pdf of the random variable X. Find the distribution function and the pdf of Y= X2Let X1, .... Xn be a random sample from a population with location pdf f(x-Q). Show that the order statistics, T(X1, ...., Xn) = (X(1), ... X(n)) are a sufficient statistics for Q and no further reduction is possible?