Question 5 - A-Level Maths

Exam Board	Edexcel
Module	S1 (Statistics 1)
Year	2013
Session	June
Topic	Discrete Probability Distributions
Type	Two unknowns from sum and expectation

A biased die with six faces is rolled. The discrete random variable \(X\) represents the score on the uppermost face. The probability distribution of \(X\) is shown in the table below.

\(x\)	1	2	3	4	5	6
\(\mathrm { P } ( X = x )\)	\(a\)	\(a\)	\(a\)	\(b\)	\(b\)	0.3

Given that \(\mathrm { E } ( X ) = 4.2\) find the value of \(a\) and the value of \(b\).
Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 20.4\)
Find \(\operatorname { Var } ( 5 - 3 X )\) A biased die with five faces is rolled. The discrete random variable \(Y\) represents the score which is uppermost. The cumulative distribution function of \(Y\) is shown in the table below.
\(y\) 1 2 3 4 5
\(\mathrm {~F} ( y )\) \(\frac { 1 } { 10 }\) \(\frac { 2 } { 10 }\) \(3 k\) \(4 k\) \(5 k\)
Find the value of \(k\).
Find the probability distribution of \(Y\). Each die is rolled once. The scores on the two dice are independent.
Find the probability that the sum of the two scores equals 2

\(y\)	1	2	3	4	5
\(\mathrm {~F} ( y )\)	\(\frac { 1 } { 10 }\)	\(\frac { 2 } { 10 }\)	\(3 k\)	\(4 k\)	\(5 k\)