Question 2 - A-Level Maths

Exam Board	Edexcel
Module	S1 (Statistics 1)
Year	2011
Session	June
Topic	Normal Distribution
Type	Find mean from probability statement

The random variable \(X \sim \mathrm {~N} \left( \mu , 5 ^ { 2 } \right)\) and \(\mathrm { P } ( X < 23 ) = 0.9192\)
1. Find the value of \(\mu\).
2. Write down the value of \(\mathrm { P } ( \mu < X < 23 )\).
3. The discrete random variable \(Y\) has probability distribution
\(y\) 1 2 3 4
\(\mathrm { P } ( Y = y )\) \(a\) \(b\) 0.3 \(c\)
where \(a , b\) and \(c\) are constants. The cumulative distribution function \(\mathrm { F } ( y )\) of \(Y\) is given in the following table
\(y\) 1 2 3 4
\(\mathrm {~F} ( y )\) 0.1 0.5 \(d\) 1.0
where \(d\) is a constant.
Find the value of \(a\), the value of \(b\), the value of \(c\) and the value of \(d\).
Find \(\mathrm { P } ( 3 Y + 2 \geqslant 8 )\).

\(y\)	1	2	3	4
\(\mathrm { P } ( Y = y )\)	\(a\)	\(b\)	0.3	\(c\)

\(y\)	1	2	3	4
\(\mathrm {~F} ( y )\)	0.1	0.5	\(d\)	1.0