Question 3 - A-Level Maths

Exam Board	OCR
Module	Further Statistics AS (Further Statistics AS)
Year	2022
Session	June
Topic	Discrete Probability Distributions
Type	Multiple unknowns from expectation and variance

3 A discrete random variable \(X\) has the following probability distribution.

\(x\)	1	2	3	4
\(\mathrm { P } ( X = x )\)	\(p\)	0.31	0.3	\(p ^ { 2 }\)

Determine the value of \(p\).
It is given that \(\mathrm { E } ( a X + b ) = \operatorname { Var } ( a X + b ) = 23.19\), where \(a\) and \(b\) are positive constants. Determine the value of \(a\) and the value of \(b\).