1. Let X have a normal distribution with parameters μ = 50 and σ2 = 144....

1. Let X have a normal distribution with parameters μ = 50 and σ2 =
144. Find the probability that X produces a value between 44 and 62. Use the
normal table A7 (be sure to show your work).

2. Let X ~ Exponential( λ ), for some fixed 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 = X1/3 (cube root of X). Using
either the CDF method or Jacobian method, show that the probability
density function of Y is given by: (10 pts)
fY(y) = 3λy^2e^(-λy3) , 0 < y < ∞.
(b) Verify that the CDF of Y is FY(y) = 1 - exp( -λy3 ), 0 < y < ∞. (5 pts)
(Hint: Use the CDF from (a) or integrate the PDF answer in (a)).
(c) If λ = 2, compute the median of Y (approximately). (5 pts)

Solutions

Expert Solution

if you are refering to the table above then the values are in the last row, ideally the table below should be the real standard normal table( biometrica format)

orchestra answered 1 year ago

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