Mathematics · Oct / Nov 2025 · Paper 61 · Question 4
YMA01
As printed in the paper
The question as text
4. A discrete random variable X has the probability distribution given in the table below, where a and b are constants. –1 0 1 2 3 x P(X = x) 0.05 0.1 a b 0.35 Given that E( X ) = 1.8
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
4(a)[3]
0.05 + 0.1 + a + b + 0.35 = 1 [a + b = 0.5]
–0.05 + a + 2b + 1.05 = 1.8 [a + 2b = 0.8]
b = 0.3 a = 0.2
[x]
–1
0
1
2
3
4(b)[2]
0.0
[F(x)]
0.15
0.35 0.65
1
5
4(c)[1]
0.15
E(5
−
3
X
)
= ×
8 0.05
+ ×
5 0.1
+
E(
X
2
)
= ×
1 0.05 1 "0.2"
+ ×
+
2 "0.2" 1 "0.3" 4 0.35[
×
−×
−×
= −
0.4]
4(d)[4]
×
+ ×
=
4 "0.3" 9 0.35[
4.6]
or E(5
−
3
X
)
=
5
−×
3 1.8[
= −
0.4]
E(5 3
−
X
)
2
=
64 0.05
×
+
25 0.1
×
+
X =
−
=
Var(
)
"4.6" 1.8
2
1.36
×
+ ×
+
×
=
4 "0.2" 1 "0.3" 16 0.35[ 12.4]
Var(5 – 3X) = 9 × “1.36”
Var(5 – 3X) = “12.4” – (“–0.4”)2
= 12.24
= 12.24
awrt 12.2
Notes
(a)
M1
for either correct equation in a and b
M1
for both correct equations in a and b
both a = 0.2 oe and b = 0.3 oe
A1
correct answers score 3 out of 3 unless from obvious
incorrect working
In part (b) and (d) working labelled in script takes priority.
Look out for working shown under the table which can be marked if it does not
contradict the working in the script.
for 3 correct values of x and their associated cumulative
(b)
B1
probabilities
for a fully correct cdf
B1ft
ft their a and b provided both are probabilities and a + b = 0.5
(c)
B1
0.15 oe
for method to find
2
with at least 3 correct (ft) nonzero
E(
X
)
products (may be implied by 4.6) only allow ft products which
use probabilities
(d)
M1
or a method to find E(5 – 3X) with at least 3 correct (ft) non
zero products (may be implied by –0.4)
or use of E(5 – 3X) = 5 – 3E(X)
for a correct method to find Var(X) ft their
E(
X
2
)
(may be
M1
implied by 1.36)
or a correct method to find E(5 – 3X)2 (may be implied by 12.4)
for use of
X ft their Var(X)
23 Var(
)
M1
or use of E(5 – 3X)2 – (E(5 – 3X))2
awrt 12.2 working must be shown Answer only send to
A1
review.
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