The random variable $X$ has probability distribution | $x$ | 1 | 2 | 3 | 4 | 5 | |-----|-----|-----|-----|-----|-----| | $P(X=x)$ | 0.10 | $p$ | 0.20 | $q$ | 0.30 | (a) Given that $E(X) = 3.5$, write down two equations involving $p$ and $q$ - Edexcel - A-Level Maths Statistics - Question 2 - 2006 - Paper 1

From the equations:

1. $p + q = 0.40$
2. $2p + 4q = 1.3$

We can express $p$ in terms of $q$ from the first equation:

$p = 0.40 - q$

Substituting this into the second equation gives:

$2(0.40 - q) + 4q = 1.3$ $0.80 - 2q + 4q = 1.3$ $2q = 0.50$ $q = 0.25$

Now substituting back to find $p$:

$p = 0.40 - 0.25 = 0.15$

To find $Var(X)$, we first need to calculate $E(X^2)$:

$E(X^2) = 1^2(0.10) + 2^2(p) + 3^2(0.20) + 4^2(q) + 5^2(0.30)$

Substituting the values of $p$ and $q$:

$E(X^2) = 0.10 + 4(0.15) + 0.60 + 16(0.25) + 25(0.30)$

Calculating each part:

• $0.10 + 0.60 + 4(0.15) + 4 + 7.5 = 14$

Now, we can calculate the variance:

$Var(X) = E(X^2) - (E(X))^2 = 14 - (3.5)^2 = 14 - 12.25 = 1.75$

The formula for the variance of a linear transformation is:

$Var(aX + b) = a^2 Var(X)$

Here, $a = -2$ and $b = 3$. Therefore:

$Var(3 - 2X) = (-2)^2 Var(X) = 4 imes 1.75 = 7.00$