Compute the first and second moments. The first moment is the same as the expected value or mean. The second moment is involved in computing the variance.
First moment:
[tex]E[X]=\displaystyle\sum_xx\,P(x)=0\cdot0.125+1\cdot0.428+\cdots+4\cdot0.083[/tex]
[tex]E[X]=1.596[/tex]
Second moment:
[tex]E[X^2]=\displaystyle\sum_xx^2\,P(x)=0^2\cdot0.125+1^2\cdot0.428+\cdots+4^2\cdot0.083[/tex]
[tex]E[X^2]=3.752[/tex]
The variance of [tex]X[/tex] is
[tex]V[X]=E[(X-E[X])^2]=E[X^2-2X\,E[X]+E[X]^2]=E[X^2]-E[X]^2[/tex]
[tex]V[X]=2.156[/tex]
The standard deviation is the square root of the variance:
[tex]\sqrt{V[X]}\approx\boxed{1.468}[/tex]
