2020 RI P2 Q9

The continuous random variable \(X\) has the distribution \(\mathrm{N}\,(\mu ,{{\sigma }^{2}})\).

It is known that \(\mathrm{P}\,(X < k)=0.2\) and \(\mathrm{P}\,(X < 7)=0.8\).

1. Show that \(\mathrm{P}\,(k < X < 7)=0.6\) and write down the value of \(\mathrm{P}\,(\mu < X < 7)\).[2]
2. Express \(\mu \) in terms of \(k\).[1]

You may use \({{\sigma }^{2}}=12\) for the rest of the question.

3. Show that \(\mu =4.0845\) correct to \(4\) decimal places.[3]
4. Find \(\mathrm{P}\,\left( \left| X \right| < k \right)\).[2]

It is given that \(2\mathrm{P}\,(X\le r)=3\mathrm{P}\,(X>r)\) for a certain constant, \(r\).

5. Ten independent observations of \(X\) are randomly selected. Find the probability that there are more observations of \(X\) with values greater than \(r\) than observations of \(X\) with values less than \(r\) in the selection.[4]