# Question 7:

## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch of pdf (correct shape, correct values at 0, 2, 3, 4, 10) | B1, B1, B1, B1dep | Values 0.2, 3, 4, 10 required; B1dep for overall shape |

## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $F(x) = 0$, $x < 0$ | B1 | Top line |
| For $0 \leq x \leq 3$: $F(x) = \int_0^x \dfrac{t^2}{45}\,dt = \left[\dfrac{t^3}{135}\right]_0^x = \dfrac{x^3}{135}$ | M1A1 | |
| For $3 < x < 4$: $F(x) = \int_3^x \dfrac{1}{5}\,dt + \dfrac{1}{5} = \left[\dfrac{t}{5}\right]_3^x + \dfrac{1}{5} = \dfrac{x}{5} - \dfrac{2}{5}$ | M1A1 | or $F(x) = \int \frac{1}{5}dx + C$ and uses $F(3) = \frac{1}{5}$ |
| For $4 \leq x \leq 10$: $F(x) = \int_4^x \!\left(\dfrac{1}{3} - \dfrac{t}{30}\right)dt + \dfrac{2}{5} = \left[\dfrac{t}{3} - \dfrac{t^2}{60}\right]_4^x + \dfrac{2}{5} = \dfrac{x}{3} - \dfrac{x^2}{60} - \dfrac{2}{3}$ | M1A1 | or $F(x) = \int\!\left(\frac{1}{3}-\frac{x}{30}\right)dx + C$ and uses $F(4)=\frac{2}{5}$ or $F(10)=1$ |
| $F(x) = 1$, $x > 10$ | B1 | Bottom line |

## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $F(8) = \dfrac{8}{3} - \dfrac{8^2}{60} - \dfrac{2}{3} = \dfrac{14}{15} = 0.933$ | M1, A1 cso | |

# Question (a) - CDF/Graph Question

## Part (a) - Graph

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Curve starting at $(0,0)$ with correct curvature | B1 | Must start at $(0,0)$ and have correct curvature |
| Horizontal line joining first section (not dotted) | B1 | Must be a solid horizontal line |
| Straight line with negative gradient joining horizontal line, stopping on positive $x$-axis | B1 | Negative gradient, terminates on positive $x$-axis |
| Fully correct graph with labels $0.2, 3, 4, 10$ in correct places | B1 | Dependent on first 3 marks being gained |

## Part (b) - CDF Integration

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Attempt to integrate $\int_0^x \frac{t^2}{45}\,dt$ | M1 | Ignore limits; need $x^n \to x^{n+1}$ |
| Attempt to integrate $\int_3^x \frac{1}{5}\,dt$ + their $F(3)$ using correct limits | M1 | Or: integrate $\int\frac{1}{5}dx + C$ substituting in $3$ and equating to their $F(3)$, or substituting in $4$ and equating to their $F(4)$ from the $4\leq x\leq 10$ line |
| Attempt to integrate $\int_4^x \frac{1}{3}-\frac{x}{30}\,dt$ + their $F(4)$ using correct limits | M1 | Or: integrate $\int\frac{1}{3}-\frac{x}{30}\,dt + C$, substituting in $4$ equating to their $F(4)$, or substituting in $10$ and putting $= 1$ |
| All correct expressions and ranges | A marks | Do not penalise $\leq$ instead of $<$ and $\geq$ instead of $>$ |

## Part (c)

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Substituting $8$ into the $4$th line of their cdf **or** $F(3)+F(4)-F(3)+F(8)-F(4)$ | M1 | Or $1-\int_8^{10}\frac{1}{3}-\frac{x}{30}\,dt$ (attempt to integrate needed); or use areas e.g. $1-\frac{1}{2}\times 2\times\frac{1}{15}$ or $1-\frac{1}{15}$ |
| $\dfrac{14}{15}$ awrt $0.933$ | A1 | From correct working. If using $F(3)+F(4)-F(3)+F(8)-F(4)$ then $F(x)$ must be correct |

---