The probability distribution of random variable X is given by:
We know that
![\[\sum\limits_{i=1}^{n}{P({{X}_{i}})=1}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6dde044c3789c324ac59130b15545b02_l3.png "Rendered by QuickLaTeX.com")
So,
![\[k\text{ }+\text{ }4k\text{ }+\text{ }9k\text{ }+\text{ }8k\text{ }+\text{ }10k\text{ }+\text{ }12k\text{ }=\text{ }1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-00c6bcc3e42d6e1a7f5edcfc27cfaca8_l3.png "Rendered by QuickLaTeX.com")
![\[44k\text{ }=\text{ }1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1ca65d74083140e4f2955f2127e82b30_l3.png "Rendered by QuickLaTeX.com")
![\[k=1/44\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-22d5060084a83b036e2254f2eef5e0ad_l3.png "Rendered by QuickLaTeX.com")
(i) E(X) =
=
![\[1\text{ }\times \text{ }k\text{ }+\text{ }2\text{ }\times \text{ }4k\text{ }+\text{ }3\text{ }\times \text{ }9k\text{ }+\text{ }4\text{ }\times \text{ }8k\text{ }+\text{ }5\text{ }\times \text{ }10k\text{ }+\text{ }6\text{ }\times \text{ }12k\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2e830bc3b9ae5f368d89dff48211233f_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }k\text{ }+\text{ }8k\text{ }+\text{ }27k\text{ }+\text{ }32k\text{ }+\text{ }50k\text{ }+\text{ }72k\text{ }=\text{ }190k\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-81ae06d9f053a39e11f2011ce9c3a85c_l3.png "Rendered by QuickLaTeX.com")
=
![\[190\text{ }\times \text{ }1/44\text{ }=\text{ }95/22\text{ }=\text{ }4.32\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4eac27d4800d0177005feb1d6aced9b3_l3.png "Rendered by QuickLaTeX.com")
(approx.)
(ii)
![\[E({{X}^{2}})\text{ }=\text{ }3\left( k\text{ }+\text{ }4\text{ }\times \text{ }4k\text{ }+\text{ }9\text{ }\times \text{ }9k\text{ }+\text{ }16\text{ }\times \text{ }8k\text{ }+\text{ }25\text{ }\times \text{ }10k\text{ }+\text{ }36\text{ }\times \text{ }12k \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3bab49fce3ef0e460bbdc5c1f2234ee5_l3.png "Rendered by QuickLaTeX.com")
=
![\[3\left( k\text{ }+\text{ }16k\text{ }+\text{ }81k\text{ }+\text{ }128k\text{ }+\text{ }250k\text{ }+\text{ }432k \right)\text{ }=\text{ }3\left( 908k \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-92cec6e5fe8a9ab3562bddc9a57f9545_l3.png "Rendered by QuickLaTeX.com")
=
![\[3\text{ }\times 908\text{ }\times 1/44\text{ }=\text{ }2724/44\text{ }=\text{ }61.9\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-cef124855f2980538de5e6b2e51b6d66_l3.png "Rendered by QuickLaTeX.com")
(approx.)
(iii)
![\[P\left( X\text{ }\ge \text{ }4 \right)\text{ }=\text{ }P\left( X\text{ }=\text{ }4 \right)\text{ }+\text{ }P\left( X\text{ }=\text{ }5 \right)\text{ }+\text{ }P\left( X\text{ }=\text{ }6 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b372c848f48fef4f46afd6241c0839f6_l3.png "Rendered by QuickLaTeX.com")
=
![\[8k\text{ }+\text{ }10k\text{ }+\text{ }12k\text{ }=\text{ }30k\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-633dab1bcc11cb688ddc14b014ead567_l3.png "Rendered by QuickLaTeX.com")
=
![\[30\text{ }\times \text{ }1/44\text{ }=\text{ }15/22\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5d8929c32d46bda3d630f762e0a5c8cc_l3.png "Rendered by QuickLaTeX.com")
The probability distribution of a random variable x is given as under: where k is a constant. Calculate (i) E(X) (ii)
![\[\mathbf{E}\text{ }(\mathbf{3}{{\mathbf{X}}^{\mathbf{2}}})\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-359841d615186c48f07fb2ecf9ffc135_l3.png "Rendered by QuickLaTeX.com")
![\[\mathbf{P}\left( \mathbf{X}\text{ }{}^\text{3}\text{ }\mathbf{4} \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bef0b1e97b70143ca1a104413a9288cb_l3.png "Rendered by QuickLaTeX.com")
