Question 6 - A-Level Maths

Exam Board	OCR MEI
Module	Further Statistics Minor (Further Statistics Minor)
Year	2024
Session	June
Topic	Discrete Probability Distributions
Type	Multiple unknowns from expectation and variance

6 The probability distribution of a discrete random variable, \(X\), is shown in the table below.

\(x\)	0	1	2
\(\mathrm { P } ( X = x )\)	\(1 - a - b\)	\(a\)	\(b\)

Find \(\mathrm { E } ( X )\) in terms of \(a\) and \(b\).
1. In the case where \(\mathrm { E } ( \mathrm { X } ) = \mathrm { a } + 0.4\), find an expression for \(\operatorname { Var } ( X )\) in terms of \(a\).
2. In this case, show that the greatest possible value of \(\operatorname { Var } ( X )\) is 0.65 . You must state the associated value of \(a\).
You are now given instead that \(\mathrm { E } ( X )\) is not known.
1. State the least possible value of \(\operatorname { Var } ( X )\).
2. Give all possible pairs of values of \(a\) and \(b\) which give the least possible value of \(\operatorname { Var } ( X )\) stated in part (c)(i).