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Additional Problem 3. If X is a continuous random variable having cdf F, then its median is defined as that value of m for which F(m) 0.5. Find the median for random variables with the following density functions (a) f(x) = e-*, x 0 (b) f(x) = 1, 0 〈 x 1. (c) f(x) 6x(1 - x),0 <1.

Answer 1

a)

as F(m)=0.5

$\int_{0}^{m}$ f(x) dx =0.5

$\int_{0}^{m}$e^-x dx =(-e^-x) |^m₀ =0.5

1-e^-m =0.5

m=-ln(0.5)=0.6931

b)

$\int_{0}^{m}$ f(x) dx =0.5

$\int_{0}^{m}$1 dx =(^_x) |^m₀ =0.5

m=0.5

c)

$\int_{0}^{m}$ f(x) dx =0.5

$\int_{0}^{m}$6(x-x²) dx =(6(x²/2-x³/3) |^m₀ =0.5

-2m³+3m²-0.5 =0

solving above: m=0.5