consider a continuous uniform random variable over 01 find E

consider a continuous uniform random variable over [0,1]. find E[x^2]-(E[x])^2

Solution

For any random variable X if E(X2 ) < , then we define it’s variance as

V (X) = E(X µ)^ 2 where µ = E(X).

Note that E(X µ)^ 2 = E(X ^2 2Xµ + µ^ 2 ) = E(X ^2 ) 2E(X)µ + E(µ ^2 ) = E(X^ 2 ) 2µ ^2 + µ^ 2

= E(X^ 2 ) µ ^2 = E(X^ 2 ) E(X)^ 2

Please note that, V (X) = E(X^2 )E(X)^ 2 can also be used as a definition of variance

So the results is variance as probability density function is not give the answer is variance of the variable