Question 7 - A-Level Maths

Exam Board	AQA
Module	S2 (Statistics 2)
Year	2008
Session	June
Topic	Discrete Probability Distributions
Type	Calculate E(X) from given distribution

The number of text messages, $N$, sent by Peter each month on his mobile phone never exceeds 40. When $0 \leqslant N \leqslant 10$, he is charged for 5 messages.
When $10 < N \leqslant 20$, he is charged for 15 messages.
When $20 < N \leqslant 30$, he is charged for 25 messages.
When $30 < N \leqslant 40$, he is charged for 35 messages.
The number of text messages, $Y$, that Peter is charged for each month has the following probability distribution:
$\boldsymbol { y }$ 5 15 25 35
$\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )$ 0.1 0.2 0.3 0.4
1. Calculate the mean and the standard deviation of $Y$.
2. The Goodtime phone company makes a total charge for text messages, $C$ pence, each month given by $$C = 10 Y + 5$$ Calculate $\mathrm { E } ( C )$.
The number of text messages, $X$, sent by Joanne each month on her mobile phone is such that $$\mathrm { E } ( X ) = 8.35 \quad \text { and } \quad \mathrm { E } \left( X ^ { 2 } \right) = 75.25$$ The Newtime phone company makes a total charge for text messages, $T$ pence, each month given by $$T = 0.4 X + 250$$ Calculate $\operatorname { Var } ( T )$.

\(\boldsymbol { y }\)	5	15	25	35
\(\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )\)	0.1	0.2	0.3	0.4