![Let ? and ?^2 denote the mean and variance of the random variable X. Determine E[((X-?))/?]
and E{[((X-?))/?]^2 } and support your answers.
Solution
u, are known constants
E[ (x-u)/ ] = E [ (x/) - (u/) ]
= E[ x/ ] - u/
= (1/) * E[x] - u/ = (1/)*u - u/
= 0
E [ ((x-u)/)2 ] = (1/2 ) * E [ X2 + u2 - 2*x*u ]
= (1/2 ) * ( E [ X2 ] + u2 - 2*E[x]*u )
= (1/2 ) * ( E [ X2 ] + u2 - 2*u2 )
= (1/2 ) * ( E [ X2 ] - u2 )
(now 2 = E[x2 ] - u2)
= (1/2 ) * 2
= 1](https://image.slidesharecdn.com/letand2denotethemeanandvarianceoftherandomvariablex-230323133008-7567ba11/85/Let-and-2-denote-the-mean-and-variance-of-the-random-variable-X-pdf-1-320.jpg)
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Let and ^2 denote the mean and variance of the random variable X..pdf
- 1. Let ? and ?^2 denote the mean and variance of the random variable X. Determine E[((X-?))/?] and E{[((X-?))/?]^2 } and support your answers. Solution u, are known constants E[ (x-u)/ ] = E [ (x/) - (u/) ] = E[ x/ ] - u/ = (1/) * E[x] - u/ = (1/)*u - u/ = 0 E [ ((x-u)/)2 ] = (1/2 ) * E [ X2 + u2 - 2*x*u ] = (1/2 ) * ( E [ X2 ] + u2 - 2*E[x]*u ) = (1/2 ) * ( E [ X2 ] + u2 - 2*u2 ) = (1/2 ) * ( E [ X2 ] - u2 ) (now 2 = E[x2 ] - u2) = (1/2 ) * 2 = 1