| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Piecewise PDF with multiple regions |
| Difficulty | Standard +0.3 This appears to be an incomplete question stem that only introduces a probability density function setup. Based on typical S2 content, complete questions usually ask students to find k (normalization constant), calculate probabilities P(a<X<b), find E(X) or Var(X), or work with cumulative distribution functions - all standard textbook exercises requiring straightforward integration and formula application rather than problem-solving insight. |
| Spec | 5.03a Continuous random variables: pdf and cdf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(X \sim B(10, 0.6)\) or \(Y \sim B(10, 0.4)\) | B1 | Writing or using \(B(10,0.6)\) or \(B(10,0.4)\) in either part (i) or (ii) |
| \(P(X=6) = (0.6)^6(0.4)^4 \frac{10!}{6!4!}\) or \(P(Y=4) = (0.4)^4(0.6)^6 \frac{10!}{6!4!}\) | M1 | Allow \(^{10}C_6\) oe |
| \(= 0.2508\) | A1 | awrt 0.251 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(X < 9) = 1-(P(X=10)+P(X=9))\) or \(P(Y < 9) = 1-P(Y \leq 1)\) | M1 | Writing or using \(1-(P(X=10)+P(X=9))\) if using \(B(10,0.6)\); or \(1-P(Y \leq 1)\) if using \(B(10,0.4)\). NB use of Poisson gains M0A0 |
| \(= 1-(0.6)^{10}-(0.6)^9(0.4)^1\frac{10!}{9!1!}\) or \(= 1-0.0464\) | ||
| \(= 0.9536\) | A1 | awrt 0.954 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(X \sim B(50, 0.6)\), \(Y \sim B(50, 0.4)\) | M1 | \(1^{st}\) M1 for writing or using either \(B(50,0.6)\) or \(B(50,0.4)\) |
| \(P(X < n) \geq 0.9\), so \(P(Y > 50-n) \geq 0.9\), i.e. \(P(Y \leq 50-n) \leq 0.1\) | M1 | \(2^{nd}\) M1: \(P(Y > 50-n) \geq 0.9\) or \(P(Y \leq 50-n) \leq 0.1\) or \(P(X < 34)\) = awrt 0.844 or \(P(X < 35)\) = awrt 0.904/0.905; or \(50-n=15\) or \(50-n=16\) or \(50-n \leq 15\) or \(50-n \leq 16\) |
| \(P(X < 34) = 0.8439\) awrt 0.844; \(P(X < 35) = 0.9045\) awrt 0.904/0.905 | ||
| \(50 - n \leq 15\), so \(n \geq 35\), thus \(n = 35\) | A1 | cao 35. Do not accept \(n \geq 35\) for final A1 |
# Question 8:
## Part (a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $X \sim B(10, 0.6)$ or $Y \sim B(10, 0.4)$ | B1 | Writing or using $B(10,0.6)$ or $B(10,0.4)$ in either part (i) or (ii) |
| $P(X=6) = (0.6)^6(0.4)^4 \frac{10!}{6!4!}$ or $P(Y=4) = (0.4)^4(0.6)^6 \frac{10!}{6!4!}$ | M1 | Allow $^{10}C_6$ oe |
| $= 0.2508$ | A1 | awrt 0.251 |
## Part (a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(X < 9) = 1-(P(X=10)+P(X=9))$ or $P(Y < 9) = 1-P(Y \leq 1)$ | M1 | Writing or using $1-(P(X=10)+P(X=9))$ if using $B(10,0.6)$; or $1-P(Y \leq 1)$ if using $B(10,0.4)$. NB use of Poisson gains M0A0 |
| $= 1-(0.6)^{10}-(0.6)^9(0.4)^1\frac{10!}{9!1!}$ or $= 1-0.0464$ | | |
| $= 0.9536$ | A1 | awrt 0.954 |
## Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $X \sim B(50, 0.6)$, $Y \sim B(50, 0.4)$ | M1 | $1^{st}$ M1 for writing or using either $B(50,0.6)$ or $B(50,0.4)$ |
| $P(X < n) \geq 0.9$, so $P(Y > 50-n) \geq 0.9$, i.e. $P(Y \leq 50-n) \leq 0.1$ | M1 | $2^{nd}$ M1: $P(Y > 50-n) \geq 0.9$ or $P(Y \leq 50-n) \leq 0.1$ or $P(X < 34)$ = awrt 0.844 or $P(X < 35)$ = awrt 0.904/0.905; or $50-n=15$ or $50-n=16$ or $50-n \leq 15$ or $50-n \leq 16$ |
| $P(X < 34) = 0.8439$ awrt 0.844; $P(X < 35) = 0.9045$ awrt 0.904/0.905 | | |
| $50 - n \leq 15$, so $n \geq 35$, thus $n = 35$ | A1 | cao 35. Do not accept $n \geq 35$ for final A1 |
---\begin{enumerate}
\item The continuous random variable $X$ has probability density function given by
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q8}}