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Chapter 6: Q. 6.29 (page 260)

A variable has a density curve whose equation isy=2xfor 0<x<1and y=0and otherwisse,

(a) Graph the denisty curve of this variable.

(b) Show that the area under this density curve to the left of any number x between 0 and 1equals x2.

What percentage of all possible observations of the variable

(c) lie between 12and 34.

(d) are atleast14.

Short Answer

Expert verified

(a) The density curve of this variable is:

(b) Thus, the area under this density curve to the left of any number x between 0 and 1equals x2.

(c) The percentagee of all possible observations that lie between 12,34is 31.25%.

(d) The percentagee of all possible observations that are atleast 14is93.75%

Step by step solution

01

Part (a) Step 1. Given Information.

A variable has a density curve whose equation isy=2x.

02

Part (a) Step 2. Graph the density function.

Graph the density function y=2x for 0<x<1andy=0.

03

Part (b) Step 1. The area between 0, 1.

From part (a),

Base, b=x

Height, h=2x

Thus the area of the triangle is given by,

A=12脳x脳2x=x2

04

Part (c) Step 1. Find the percentage.

Area between 12,34=[Area to the left of 34]-[Area to the left of 12]

=[34]2-[12]2 =916-14=516=0.3125

The percentagee of all possible observations that lie between 12,34is31.25%.

05

Part (d) Step 1. Find the percentage.

[Area to the right of 14]=1-[Area to the left of 14]

localid="1652597970533" =1-(14)2=1-116=1516=0.9375

The percentagee of all possible observations that lie atleast14 is93.75%

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