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

To find the values of p and q, we can solve the system of equations derived:

1. From the first equation, express q in terms of p: $q = 0.40 - p$

2. Substitute q into the second equation: $2p + 4(0.40 - p) = 1.3$ Expanding this gives: $2p + 1.6 - 4p = 1.3$ Combining like terms results in: $-2p + 1.6 = 1.3$ Therefore: $-2p = 1.3 - 1.6\ -2p = -0.3\ p = 0.15$

3. Substitute the value of p back into the equation for q: $q = 0.40 - 0.15 = 0.25$

Thus, the values are: $p = 0.15, q = 0.25.$

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

1. Compute 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 found values for p and q: $E(X^2) = 0.10 + 4 * 0.15 + 0.60 + 16 * 0.25 + 7.5$ Simplifying gives: $E(X^2) = 0.10 + 0.60 + 0.60 + 4 + 7.5 = 14$

2. Now, calculate Var(X): $Var(X) = E(X^2) - (E(X))^2$ Thus: $Var(X) = 14 - (3.5)^2 = 14 - 12.25 = 1.75$

To find the variance of the transformed variable 3 − 2X, we can use the formula for variance under linear transformations:

1. The formula is given by: $Var(aX + b) = a^2Var(X)$ In this scenario:

   • a = -2 (since we have -2X)
   • b = 3 (which does not affect variance)

2. Therefore: $Var(3 - 2X) = (-2)^2 Var(X) = 4 * 1.75 = 7$