9709 · 6.2
Linear combinations of random variables flashcards
Revision flashcards for Cambridge 9709 Linear combinations of random variables (syllabus 6.2). Flip, recall, then mark a real past-paper question.
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What is the formula for the expectation of a linear combination, E(aX + bY)?
E(aX + bY) = aE(X) + bE(Y). This holds true whether X and Y are independent or not.
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What is the formula for the variance of a linear combination, Var(aX + bY)?
Var(aX + bY) = a²Var(X) + b²Var(Y). This formula requires X and Y to be independent random variables.
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Why are the coefficients squared (a² and b²) in the variance formula?
Because variance is measured in squared units. When a variable X is scaled by a factor 'a', its variance, which is related to the average squared deviation, is scaled by a².
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What is the formula for Var(X - Y) for independent variables X and Y?
Var(X - Y) = Var(X) + Var(Y). This is a common trap! Variances always add because uncertainty increases regardless of whether you are adding or subtracting the variables.
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If X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²) are independent, what is the distribution of X + Y?
X + Y ~ N(μ₁ + μ₂, σ₁² + σ₂²). A linear combination of independent normal variables is also a normal variable.
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What is the distribution of aX for X ~ N(μ, σ²)?
aX ~ N(aμ, a²σ²). The mean is multiplied by 'a' and the variance by 'a²'.
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What is the distribution of X + c for X ~ N(μ, σ²), where c is a constant?
X + c ~ N(μ + c, σ²). Adding a constant shifts the mean but does not change the variance (spread).
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What is the key condition required to use the formula Var(aX + bY) = a²Var(X) + b²Var(Y)?
The random variables X and Y must be independent.
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Let X₁, X₂, ..., Xₙ be a random sample of size n from a distribution with mean μ and variance σ². What are E(∑Xᵢ) and Var(∑Xᵢ)?
E(∑Xᵢ) = nμ and Var(∑Xᵢ) = nσ². This is a special case of a linear combination where all coefficients are 1.
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What is the difference between 2X and X₁ + X₂?
2X represents taking one observation and doubling it. Var(2X) = 4Var(X). X₁ + X₂ represents taking two independent observations and adding them. Var(X₁ + X₂) = Var(X₁) + Var(X₂) = 2Var(X).