Probability distributions — Edexcel A-Level Statistics
Test yourself on Probability distributions with PEARSON EDEXCEL A-Level practice questions.
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Probability distributions explained
This criterion requires you to find the mean and variance of a linear combination of independent random variables.
Read the full explanation
If each Xi has mean µi and variance σi², and the Xi are independent, then the combination ∑ ai Xi has mean ∑ ai µi and variance ∑ ai² σi². The coefficients ai are constants, so they scale the mean directly but are squared when scaling the variance. For example, if X₁ has mean 10 and variance 4, and X₂ has mean 5 and variance 9, with X₁ and X₂ independent, then 3X₁ − 2X₂ has mean 3(10) − 2(5) = 20 and variance 3²(4) + (−2)²(9) = 36 + 36 = 72. Notice that subtracting a variable still adds its variance because variance measures spread, and the negative coefficient is squared. You must also recognise when variables are not independent, in which case this formula does not apply without covariance terms.
Your focus
- State the mean and variance formulae for a linear combination of independent random variables.
- Calculate the mean and variance of a given linear combination using the correct coefficients.
- Explain why independence is necessary for the variance formula to apply.
Probability distributions exam tips
Marking Points
- Correctly identify the mean of a linear combination as ∑ ai µi, applying each coefficient to its variable's mean.
- Correctly identify the variance of a linear combination as ∑ ai² σi², squaring each coefficient and adding variances.
- Recognise that independence is required for the variance formula to hold without additional covariance terms.
- Apply the formulae accurately to numerical values, including negative coefficients, and interpret the result in context.
Examiner Tips
- 💡Write down the mean and variance of each variable before combining them to avoid arithmetic slips.
- 💡Check that the variables are independent; if not, the simple variance formula does not apply.
- 💡Use the squared coefficient for every variance term, even when the coefficient is negative.
Common Mistakes
- Forgetting to square the coefficients when calculating variance; correct by using ai² for each term.
- Subtracting variances when the combination involves subtraction; correct by adding variances because variance is always non-negative and spread adds for independent variables.
- Applying the formula to dependent variables; correct by checking independence first, as the formula requires independent random variables.