Standard +0.8 This question requires understanding of discrete uniform distributions, applying linearity of expectation and variance scaling properties, then solving simultaneous equations involving the formulas for mean and variance of consecutive integers. While the concepts are standard Further Maths material, the algebraic manipulation and systematic approach needed across multiple steps makes it moderately challenging, above average difficulty but not requiring deep insight.
The random variable \(X\) is equally likely to take any of the \(n\) integer values from \(m + 1\) to \(m + n\) inclusive. It is given that \(\text{E}(3X) = 30\) and \(\text{Var}(3X) = 36\).
Determine the value of \(m\) and the value of \(n\). [7]
Question 4:
4 | 3E(X) = 30 or E(X) = 10
9×Var(X) = 36 or Var(X) = 4
1 (n2 −1)=4
12
⇒ n = 7
E(X – m) = 1(n+1)
2 | B1
B1
M1
A1
M1 | 2.2a
2.2a
1.1
2.2a
3.1b | Used, stated or implied
One of these, used, stated or implied
Use variance of uniform
n = 7 only, no need for “reject –7”
Use expectation of uniform, e.g.
2m + n + 1 = 20. | Allow if E(3X + m) used
rather than E[3(X + m )]
OR: | Var(Y + m) = 1 (n2−1)
12 | M1
A1
M1 | 1.1
2.2a
3.1b
n = 7 only, no need for “reject –7”
⇒ n = 7
Use expectation of uniform, e.g.
E(Y + m) = ½(n + 1) + m
2m + n + 1 = 20.
10 – m = 4
m = 6 | M1
A1
[7] | 2.1
2.2a | Validly derive single equation for m
m = 6 only | NB: Var = (n – 1)2/12 is
from continuous uniform!
M1
A1
M1
1.1
2.2a
3.1b
Question | Answer | Marks | AO | Guidance
The random variable $X$ is equally likely to take any of the $n$ integer values from $m + 1$ to $m + n$ inclusive. It is given that $\text{E}(3X) = 30$ and $\text{Var}(3X) = 36$.
Determine the value of $m$ and the value of $n$. [7]
\hfill \mbox{\textit{OCR Further Statistics 2020 Q4 [7]}}