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E(aX + b) and Var(aX + b)

E(aX + bY) and Var(aX + bY) for independent X and Y

Linear combination of normal random variables is normal

Sum of independent Poisson random variables is Poisson

189 questions

Mathematics/Paper 6/Linear Combinations of Random Variables

CAIEA-Level9709-a · Paper 6

Linear Combinations of Random Variables

189 questions· page 1 of 19

Q12025 Feb/Mar·P625MMedium-EasyLinear Combinations of Random Variables

The random variables $X$ and $Y$ have the independent distributions $N(44, 16)$ and $N(30, 9)$ respectively.

Find $P(X - Y < 15)$ .

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Q42025 May/Jun·P612 partsMedium-EasyLinear Combinations of Random Variables

(a)

Find the mean and variance of the amount people pay for using the video game.

(b)

Each day, $35$ people independently use the video game.

Find the mean and variance of the total amount paid by $35$ people.

Q52025 May/Jun·P634MMediumLinear Combinations of Random Variables

In Urberia, the masses, in kilograms, of men have the distribution $N(70.3, 5.9^2)$ . A certain footbridge in Urberia can take a maximum safe load of 1500 kg. When $n$ men stand on the bridge, the probability that the bridge is unsafe is less than 0.01.

Stating a necessary assumption, find the maximum value of $n$ .

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Q12025 May/Jun·P652 partsMedium-EasyThe Poisson Distribution

(a)

Find $P(S \le 3)$ .

(b)

Show that the exact value of $\frac{P(S \le 2)}{P(S \le 1)}$ is $\frac{125}{44}$ .

Q52025 May/Jun·P655MMediumLinear Combinations of Random Variables

Candidates for a certain diploma take two tests. Their marks for the first test and the second test are modelled by the independent variables with distributions $N(38.1, 3.8^2)$ and $N(64.0, 6.1^2)$ respectively. The final mark, $F$ , for each candidate is found by doubling the mark in the first test and adding the result to the mark in the second test.

Find the probability that the mean, $\bar{F}$ , of the final marks of a random sample of 25 candidates is greater than 143.

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Q62025 Oct/Nov·P612 partsMedium-EasyLinear Combinations of Random Variables

(a)

Find the probability that the total mass of a randomly chosen large bag of potatoes and a randomly chosen small bag of potatoes is more than 3.55 kg.

(b)

Find the probability that the mass of a randomly chosen large bag of potatoes is less than 3 times the mass of a randomly chosen small bag of potatoes.

Q22025 Oct/Nov·P622 partsMedium-EasyThe Poisson Distribution

(a)

Find the standard deviation of $X + Y$ .

(b)

Find the standard deviation of $5X - Y$ .

Q62025 Oct/Nov·P632 partsMedium-EasyLinear Combinations of Random Variables

(a)

Find the probability that the total mass of a randomly chosen large bag of potatoes and a randomly chosen small bag of potatoes is more than 3.55kg.

(b)

Find the probability that the mass of a randomly chosen large bag of potatoes is less than 3 times the mass of a randomly chosen small bag of potatoes.

Q52025 Oct/Nov·P652 partsMediumLinear Combinations of Random Variables

(a)

Find $\mathrm{P}(L > S + 0.5)$ .

(b)

The price of tomatoes is $4.30 per kg.

A large bag of tomatoes and a small bag of tomatoes are chosen at random. Find the probability that the total price of the tomatoes in the two bags is less than $16.

Q42024 Feb/Mar·P622 partsMediumLinear Combinations of Random Variables

(a)

Find the probability that in a randomly chosen year the firm uses more gasoline than diesel fuel.

(b)

The costs per litre of gasoline and diesel fuel are $0.80 and $0.85 respectively.

Find the probability that the total cost of gasoline and diesel fuel in a randomly chosen year is between $20,000 and $22,000.

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