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Using jacobian method finding the probability density function
Let X ~ Exponential (λ), for some constant λ > 0. That is,fX(x) = λ e-λx = λ exp( -λx ), x > 0, ( fX(x) = 0 otherwise)
(a) Create a transformed random variable Y = ln(X) (natural log of X). Using either the CDF method or Jacobian method, show that the probability density function of Y is given by:fY(y) = λ exp( y - λey ), -∞ ≤ y ≤ ∞.
(b) Verify that the CDF of Y is FY(y) = 1 - exp (-λey ), -∞ ≤ y ≤ ∞.
(c) If λ = 1, compute the median of Y.
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The critical value of F at 95% confidence when there is a sample size of 21 for the sample with the smaller variance, and there is a sample size of 9 for the sample with the larger sample variance is:
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