A Level · Mathematics 9709 · Probability & Statistics 2

Linear combinations of random variables: A Level Mathematics questions with answers

Free Linear combinations of random variables questions for A Level Mathematics (9709), part of Probability & Statistics 2. Each question has the answer and the marking points an examiner would look for.

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Linear combinations of random variables practice questions

Question 1Structured · 9 marksA* challenge

The masses of apples are X ~ N(50, 16) grams and the masses of oranges are Y ~ N(77, 9) grams, independently.

(a)

Find the probability that the total mass of three randomly chosen apples is more than 65 g greater than the mass of one randomly chosen orange.

[4]
(b)

Find the probability that three times the mass of one apple is more than twice the mass of one orange.

[4]
(c)

Explain why the answer to (b) is not the same as the probability that the total mass of three apples is more than the total mass of two oranges.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.8553
E(X1 + X2 + X3 − Y) = 73 B1
Var = 57 B1
standardise M1
0.8553 A1
4
(b)0.3828
E(3X − 2Y) = -4 B1
Var(3X − 2Y) = 9 × 16 + 4 × 9 = 180 B1
standardise and correct tail M1
0.3828 A1
4
(c)3X uses one apple tripled (variance 9σ2); three different apples have variance 3σ2
variance of 3X is 9Var(X) but of X1 + X2 + X3 is 3Var(X) B1
1
Question 2Structured · 6 marks

X ~ N(220, 82) and Y ~ N(100, 42) are independent.

Find P(X + 2Y > 431) and explain why this is different from P(X + Y1 + Y2 > 431), where Y1, Y2 are independent observations of Y.

[6]
Show answers and mark scheme
PartAnswer and marking pointsMarks
0.165
E = 420 B1
Var = 128 B1
standardise M1
0.165 A1
Var(X + Y1 + Y2) = 96 B1
so the probabilities differ B1
6
Question 3Structured · 4 marks

X ~ N(52, 4) and Y ~ N(35, 25) are independent. Find P(X + Y > 91.8).

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
0.184
E = 87 B1
Var = 29 B1
Standardise M1
0.184 A1
4
Question 4Structured · 9 marksA* challenge

The masses of apples are X ~ N(46, 9) grams and the masses of oranges are Y ~ N(72, 16) grams, independently.

(a)

Find the probability that the total mass of three randomly chosen apples is more than 69 g greater than the mass of one randomly chosen orange.

[4]
(b)

Find the probability that three times the mass of one apple is more than twice the mass of one orange.

[4]
(c)

Explain why the answer to (b) is not the same as the probability that the total mass of three apples is more than the total mass of two oranges.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.3237
E(X1 + X2 + X3 − Y) = 66 B1
Var = 43 B1
standardise M1
0.3237 A1
4
(b)0.3091
E(3X − 2Y) = -6 B1
Var(3X − 2Y) = 9 × 9 + 4 × 16 = 145 B1
standardise and correct tail M1
0.3091 A1
4
(c)3X uses one apple tripled (variance 9σ2); three different apples have variance 3σ2
variance of 3X is 9Var(X) but of X1 + X2 + X3 is 3Var(X) B1
1
Question 5Structured · 6 marks

X ~ N(239, 102) and Y ~ N(132, 52) are independent.

Find P(X + 2Y > 517) and explain why this is different from P(X + Y1 + Y2 > 517), where Y1, Y2 are independent observations of Y.

[6]
Show answers and mark scheme
PartAnswer and marking pointsMarks
0.161
E = 503 B1
Var = 200 B1
standardise M1
0.161 A1
Var(X + Y1 + Y2) = 150 B1
so the probabilities differ B1
6
Question 6Structured · 4 marks

X ~ N(55, 4) and Y ~ N(32, 25) are independent. Find P(X + Y > 90.2).

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
0.274
E = 87 B1
Var = 29 B1
Standardise M1
0.274 A1
4

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