Given,
![\[X\text{ }=\text{ }0,\text{ }1,\text{ }2,\text{ }3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-44e46a23ba750709b0ef00986e4eb82d_l3.png "Rendered by QuickLaTeX.com")
P(X = r)
![\[={{~}^{n}}{{C}_{r}}~{{p}^{r}}~{{q}^{n-r}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-15c49d4cad9e6252198036e1661d110f_l3.png "Rendered by QuickLaTeX.com")
Where
![\[n\text{ }=\text{ }3,\text{ }p\text{ }=\text{ 1/2},\text{ }q\text{ }=\text{ 1/2}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f2f2d15c4099494bdece3bc059dbffe8_l3.png "Rendered by QuickLaTeX.com")
and
![\[r\text{ }=\text{ }0,\text{ }1,\text{ }2,\text{ }3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ea1afbfe59348abbfe188b159edd9292_l3.png "Rendered by QuickLaTeX.com")
![\[P\left( X\text{ }=\text{ }0 \right)\text{ }=\text{ 1/2}\times \text{1/2}\times \text{1/2 }=\text{ }1/8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d00efc0a6304b422d666867844de1f65_l3.png "Rendered by QuickLaTeX.com")
![\[P\left( X\text{ }=\text{ }1 \right)\text{ }=\text{ }3\text{ }\times \text{1/2}\times \text{ 1/2 }\times \text{1/2 }=\text{ }3/8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e78796e32d685e023ff3a65939719816_l3.png "Rendered by QuickLaTeX.com")
![\[P\left( X\text{ }=\text{ }2 \right)\text{ }=\text{ }3\text{ }\times \text{ 1/2}\times 1/2\times 1/2\text{ }=\text{ }3/8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-51a23c19facc3127dc69b65b59366b6c_l3.png "Rendered by QuickLaTeX.com")
![\[P\left( X\text{ }=\text{ }3 \right)\text{ }=\text{ 1/2}\times \text{1/2}\times \text{1/2 }=\text{ }3/8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6e27f33edd9fc778c4d321fd961621b6_l3.png "Rendered by QuickLaTeX.com")
The probability distribution table is:
Now,
![\[E\left( X \right)\text{ }=\text{ }0\text{ }+\text{ }1\text{ }\times 3/8\text{ }+\text{ }2\text{ }\times 3/8\text{ }+\text{ }3\text{ }\times \text{ }1/8\text{ }=\text{ }3/8\text{ }+\text{ }6/8\text{ }+\text{ }3/8\text{ }=\text{ }12/8\text{ }=\text{ }3/2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bfbc046cf1ea1f1b4fb54ccc22f0119b_l3.png "Rendered by QuickLaTeX.com")
![\[E({{X}^{2}})\text{ }=\text{ }0\text{ }+\text{ }1\text{ }\times 3/8\text{ }+\text{ }4\text{ }\times 3/8\text{ }+\text{ }9\text{ }\times 1/8\text{ }=\text{ }3/8\text{ }+\text{ }12/8\text{ }+\text{ }9/8\text{ }=\text{ }24/8\text{ }=\text{ }3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0a4c2e59544cb5392e08d39b29b0e188_l3.png "Rendered by QuickLaTeX.com")
W.k.t,
![\[Var\left( X \right)\text{ }=\text{ }E({{X}^{2}})\text{ }\text{ }{{\left[ E\left( X \right) \right]}^{2}}~=\text{ }3\text{ }\text{ }{{\left( 3/2 \right)}^{2}}~=\text{ }3\text{ }\text{ }{{\left( 3/2 \right)}^{2}}~=\text{ }3\text{ }\text{ }9/4\text{ }=\text{ 3/4}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-740f940f4a035a22282728f0e1b82035_l3.png "Rendered by QuickLaTeX.com")
Thus, the standard deviation = Var(X) =
![\[\sqrt{3/4}\text{ }=\text{ }\sqrt{3}/2\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ab46f9c1a253c04374c7f03804df0425_l3.png "Rendered by QuickLaTeX.com")
Given,
P(X = r)
Where
and
The probability distribution table is:
Now,
W.k.t,
Thus, the standard deviation = Var(X) =
