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The PDF for the t distribution with q degrees of freedom is T((q+1)/2) |'(q/2)/#q (1+) f (x; q) = Cf. equation (3.6). Using properties of the exponential function, and the result that r( ++) - /2= (})' exp(-) as q→ o, prove that f (x; q) tends to the PDF of an N(0, 1) RV in this limit. Hint: Write -1/2-1/24 (1+*) Derive the PDF of variable Y=Z, where Z is N(0, 1). The PDF for the F-distribution with (1. q) degrees of freedom is -(+1)/2 r(q+1)/2) 8 (x; q) = I(q/2)/#q 0 >r>0 Using the above limiting results, show that f (x: q) tends to the PDF of Y as q-o.

The PDF for the t distribution with q degrees of freedom is T((q+1)/2) |'(q/2)/#q (1+) f (x; q) = Cf. equation (3.6). Using properties of the exponential function, and the result that r( ++) - /2= (})' exp(-) as q→ o, prove that f (x; q) tends to the PDF of an N(0, 1) RV in this limit. Hint: Write -1/2-1/24 (1+*) Derive the PDF of variable Y=Z, where Z is N(0, 1). The PDF for the F-distribution with (1. q) degrees of freedom is -(+1)/2 r(q+1)/2) 8 (x; q) = I(q/2)/#q 0 >r>0 Using the above limiting results, show that f (x: q) tends to the PDF of Y as q-o.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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The PDF for the t distribution with q degrees of freedom is -(4+1)/2 T((q+1)/2) T'(q/2) /#q (1+) f (x; q) = Cf. equation (3.6). Using properties of the exponential function, and the result that esp(-") еxp as q→ o, prove that f (x; q) tends to the PDF of an N(0, 1) RV in this limit. Hint: Write -(4+1)/2 -1/2-1/24 (1+) Derive the PDF of variable Y = Z', where Z is N(0, 1). The PDF for the F-distribution with (1. q) degrees of freedom is -(4+1)/2 T((q+1)/2) g (x; q) = T(q/2)S#q .0 <x<x, Using the above limiting results, show that f (x; q) tends to the PDF of Y as q→.
Transcribed Image Text:The PDF for the t distribution with q degrees of freedom is -(4+1)/2 T((q+1)/2) T'(q/2) /#q (1+) f (x; q) = Cf. equation (3.6). Using properties of the exponential function, and the result that esp(-") еxp as q→ o, prove that f (x; q) tends to the PDF of an N(0, 1) RV in this limit. Hint: Write -(4+1)/2 -1/2-1/24 (1+) Derive the PDF of variable Y = Z', where Z is N(0, 1). The PDF for the F-distribution with (1. q) degrees of freedom is -(4+1)/2 T((q+1)/2) g (x; q) = T(q/2)S#q .0 <x<x, Using the above limiting results, show that f (x; q) tends to the PDF of Y as q→.
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