# Question 5:

## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| The distribution is symmetric about the value 2 | B1 cso | For argument using symmetry "distribution is symmetric" B1. "probs are symmetric" B0. "it is symmetric" is B0. Or a correct expression $(6a+4b)$ and use of sum of probs $=1$ |

## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Sum of probs $= 1$ (or use of $E(X)=2$) leading to $3a + 2b = 1$ | B1 | For $3a+2b=1$ (o.e.) (any equivalent correct equation, needn't be simplified) |

## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(X^2) = (-1)^2b + 2^2a + 4^2a + 5^2b \ [= 20a+26b \text{ ...condone } 24b]$ | M1 | For a full expression for $E(X^2)$. Condone $-1^2b$... or $20a+26b$ or $20a+24b$. Allow $\text{Var}(X)$ called $E(X^2)$. M0 for $\frac{20a+26b}{5}$ unless you see $E(X^2)=20a+26b$ (o.e.) first |
| $7.1 = 20a + \text{``}26\text{''}b - 2^2$ or $7.1 = 20a + \text{``}26\text{''}b - (6a+4b)^2$ or $7.1 = 8a+18b$ | M1 | For use of the correct formula to form an equation for $a$ and $b$, ft their $E(X^2)$ |
| $11.1 = 20a + 26b$ | A1 | For $11.1 = 20a+26b$ (or equivalent but must be only 3 non-zero terms) |

## Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| e.g. (b)$\times13$ and subtract (c) yielding: $1.9 = 19a$ | M1 | For solving their 2 linear equations in $a$ and $b$ and reducing to an equation in one variable. Condone 1 arithmetic or sign error |
| $a = \mathbf{0.1}$ and $b = \mathbf{0.35}$ | A1, A1 | For $a=0.10$ or exact equivalent. For $b=0.35$ or exact equivalent. One correct value scores M1 and the relevant A1 and both correct scores 3/3 |

## Part (e)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $[E(Y) = 10 - 3E(X) = 10-3\times2] = \mathbf{4}$ | B1 | |

## Part (e)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $[\text{Var}(Y)] = (-3)^2\text{Var}(X)$ | M1 | For correct use of the $\text{Var}(aX+b)$ formula. Condone $-3^2$ if it later becomes $+9$. Or $[E(Y^2)] = 79.9$ and $[\text{Var}(Y)] = 79.9 -$ their $(E(Y))^2$ |
| $= \mathbf{63.9}$ | A1 | For 63.9 |

## Part (f):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $Y > X$ gives: $10-3X > X$ leading to $10 > 3X+X$ or $Y > 2.5$ or $X < 2.5$ | M1 | For an attempt to solve the linear inequality leading to $10>3X+X$ or $Y>2.5$ or $Y\geqslant4$ |
| $X < 2.5$ means $X = -1$, 0 and 2 | A1 | For the correct 3 values of $X$ or prob. dist. for $Y$ and $y=4,10,13$ or $P(X<2.5)=2a+b$ |
| $P(Y>X) = 2a+b = \mathbf{0.55}$ or $\frac{11}{20}$ (o.e.) | A1ft | For an answer $=$ their $2a+b$ provided $a$, $b$ and $2a+b$ are probabilities. Must be a value. BUT $2a+b$ only is M0 |

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