11. Let X be a continuous random variable with density function fare-102 for 10 f(1) =...

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Question

11. Let X be a continuous random variable with density function fare-102 for 10 f(1) = lo otherwise where a > 0. What is the

Answer 1

Answer #1

here for this to be valid :

$\int_{0}^{\infty }$ f(x) dx must be 1

$\int_{0}^{\infty }$ f(x) dx = $\int_{0}^{\infty }$ ax² e^-10x dx =a*(-x²e^-10x/10-2xe^-10x/100-2e^-10x/1000) |₀ =a/500 =1

a =500

for (d/dx)f(x) =(d/dx)*(500x²e^-10x) =500*(2x*e^-10x-10x²e^-10x)

putting above equal to 0 ,

500*(2x*e^-10x-10x²e^-10x) =0

x =2/10 =0.2 =mode

therefore P(X>=mode )=P(X>=0.2) =1-P(X<=0.2)

=1- f(x) dx =1-500x² e^-10x dx =1-500*(-x²e^-10x/10-2xe^-10x/100-2e^-10x/1000) |^0.2₀ =1-0.3233

=0.6767

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Answer 2

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