# Question: Suppose X is an exponential random variable with PDF fX

Suppose X is an exponential random variable with PDF, fX (x) = exp (– x) u (x). Find a transformation, Y= g(X) so that the new random variable Y has a Cauchy PDF given by

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Suppose a random variable X has a PDF which is nonzero only on the interval [ 0, 8 ) . That is, the random variable cannot take on negative values. Prove that An Erlang Random variable has a PDF of the form (a) Find the characteristic function, ϕX( ω) . (b) Find the Taylor series expansion of ϕX( ω) . (c) Find a general expression for the k th moment of X. Derive a relationship between the k th factorial moment for a nonnegative, integer-valued random variable and the coefficients of the Taylor series expansion of its probability- generating function, HX( z) , about the point ...A random variable has a moment- generating function given by (a) Find the PDF of the random variable. (b) Use the moment- generating function to find an expression for the k th moment of the random variable. Let X be an Erlang random variable with PDF, Derive a saddle point approximation for the left tail probability, Pr (X< xo). Compare your result with the exact value for 0 ≤ xo < E [X].Post your question