Question: Let X ∼ Expo(λ). You can assume you know that λX ∼ Expo(1), and that the nth moment of an Expo(1)random variable is n!. Find the skewness of X.
This is a question for class. My prof has given me some guidance but I can't wrap my mind around it.
My prof said I should start by considering the equation for skewness very generically, and use the moment generating function. He also gave me a demonstration of swapping the interior of the skewness equation:
Skew(X) = [FONT=MathJax_Main](([FONT=MathJax_Math-italic]X[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]μ)/[FONT=MathJax_Math-italic]σ[FONT=MathJax_Main])^[FONT=MathJax_Main]3 [FONT=MathJax_Main]= [/FONT][FONT=MathJax_Main](([/FONT][FONT=MathJax_Math-italic]X[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]E[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main]))/Sqrt([/FONT][/FONT][/FONT][FONT=MathJax_Math-italic]V[FONT=MathJax_Math-italic]a[FONT=MathJax_Math-italic]r[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[FONT=MathJax_Main]))[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][FONT=MathJax_Main])^[FONT=MathJax_Main]3[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT]
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Then I know some properties of an exponential distribution: [FONT=MathJax_Math-italic]E[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[FONT=MathJax_Main])[FONT=MathJax_Main]=[FONT=MathJax_Main]1/[FONT=MathJax_Math-italic]λ[FONT=MathJax_Main], [/FONT][FONT=MathJax_Math-italic]V[/FONT][FONT=MathJax_Math-italic]a[/FONT][FONT=MathJax_Math-italic]r[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1/[FONT=MathJax_Math-italic]λ^[FONT=MathJax_Main]2[/FONT][/FONT][/FONT][FONT=MathJax_Main], [/FONT][FONT=MathJax_Math-italic]S[/FONT][FONT=MathJax_Math-italic]D[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1/[FONT=MathJax_Math-italic]λ,
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So with that I can say that: ([FONT=MathJax_Math-italic]X[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]E[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[FONT=MathJax_Main]))/[FONT=MathJax_Size2]√[FONT=MathJax_Math-italic]V[FONT=MathJax_Math-italic]a[FONT=MathJax_Math-italic]r[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main]) [/FONT][/FONT][/FONT][/FONT][/FONT][FONT=MathJax_Main]= ([/FONT][FONT=MathJax_Math-italic]x[FONT=MathJax_Main]−[FONT=MathJax_Main]1/[FONT=MathJax_Math-italic]λ)/([/FONT][/FONT][FONT=MathJax_Main]1/[FONT=MathJax_Math-italic]λ) [/FONT][/FONT][/FONT][/FONT][FONT=MathJax_Main]= [/FONT][FONT=MathJax_Math-italic]λ[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1
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therefore, [FONT=MathJax_Math-italic]S[FONT=MathJax_Math-italic]k[FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]w[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[FONT=MathJax_Main])[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]λ[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])^[FONT=MathJax_Main]3
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But this is where I get stuck. My first problem is that the textbook gives the equation for Skew as: [FONT=MathJax_Math-italic]S[FONT=MathJax_Math-italic]k[FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]w[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[FONT=MathJax_Main])[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]E[/FONT][FONT=MathJax_Main](([/FONT][FONT=MathJax_Math-italic]X[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]μ)/[/FONT][FONT=MathJax_Math-italic]σ[/FONT][/FONT][/FONT][FONT=MathJax_Main])^[FONT=MathJax_Main]3[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT]
which is not what my prof wrote and I'm not sure how I can just ignore the expected value. Second, I don't know how to use the MGF in this approach, or how to use the fact that the nth moment = n!
This is a question for class. My prof has given me some guidance but I can't wrap my mind around it.
My prof said I should start by considering the equation for skewness very generically, and use the moment generating function. He also gave me a demonstration of swapping the interior of the skewness equation:
Skew(X) = [FONT=MathJax_Main](([FONT=MathJax_Math-italic]X[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]μ)/[FONT=MathJax_Math-italic]σ[FONT=MathJax_Main])^[FONT=MathJax_Main]3 [FONT=MathJax_Main]= [/FONT][FONT=MathJax_Main](([/FONT][FONT=MathJax_Math-italic]X[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]E[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main]))/Sqrt([/FONT][/FONT][/FONT][FONT=MathJax_Math-italic]V[FONT=MathJax_Math-italic]a[FONT=MathJax_Math-italic]r[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[FONT=MathJax_Main]))[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][FONT=MathJax_Main])^[FONT=MathJax_Main]3[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT]
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Then I know some properties of an exponential distribution: [FONT=MathJax_Math-italic]E[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[FONT=MathJax_Main])[FONT=MathJax_Main]=[FONT=MathJax_Main]1/[FONT=MathJax_Math-italic]λ[FONT=MathJax_Main], [/FONT][FONT=MathJax_Math-italic]V[/FONT][FONT=MathJax_Math-italic]a[/FONT][FONT=MathJax_Math-italic]r[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1/[FONT=MathJax_Math-italic]λ^[FONT=MathJax_Main]2[/FONT][/FONT][/FONT][FONT=MathJax_Main], [/FONT][FONT=MathJax_Math-italic]S[/FONT][FONT=MathJax_Math-italic]D[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1/[FONT=MathJax_Math-italic]λ,
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So with that I can say that: ([FONT=MathJax_Math-italic]X[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]E[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[FONT=MathJax_Main]))/[FONT=MathJax_Size2]√[FONT=MathJax_Math-italic]V[FONT=MathJax_Math-italic]a[FONT=MathJax_Math-italic]r[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[/FONT][FONT=MathJax_Main]) [/FONT][/FONT][/FONT][/FONT][/FONT][FONT=MathJax_Main]= ([/FONT][FONT=MathJax_Math-italic]x[FONT=MathJax_Main]−[FONT=MathJax_Main]1/[FONT=MathJax_Math-italic]λ)/([/FONT][/FONT][FONT=MathJax_Main]1/[FONT=MathJax_Math-italic]λ) [/FONT][/FONT][/FONT][/FONT][FONT=MathJax_Main]= [/FONT][FONT=MathJax_Math-italic]λ[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1
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therefore, [FONT=MathJax_Math-italic]S[FONT=MathJax_Math-italic]k[FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]w[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[FONT=MathJax_Main])[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]λ[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])^[FONT=MathJax_Main]3
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But this is where I get stuck. My first problem is that the textbook gives the equation for Skew as: [FONT=MathJax_Math-italic]S[FONT=MathJax_Math-italic]k[FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]w[FONT=MathJax_Main]([FONT=MathJax_Math-italic]X[FONT=MathJax_Main])[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]E[/FONT][FONT=MathJax_Main](([/FONT][FONT=MathJax_Math-italic]X[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]μ)/[/FONT][FONT=MathJax_Math-italic]σ[/FONT][/FONT][/FONT][FONT=MathJax_Main])^[FONT=MathJax_Main]3[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT]
which is not what my prof wrote and I'm not sure how I can just ignore the expected value. Second, I don't know how to use the MGF in this approach, or how to use the fact that the nth moment = n!