Continuous random variables in A Level Math Paper 6 (Statistics 2): common mistakes and examiner advice

Continuous random variable questions are mostly calculus: integrate to find a constant, a probability, the median, the mean and the variance. The reports show that the calculus is usually sound. The marks go missing in the limits, the signs and the details of the setup. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• Setting up the equation. Not equating the integral to 1 clearly, setting it equal to the wrong value (0.25 or 1 in one question, and 1 where the right value was 117/256), or leaving out constants and factors such as 8/7 or 24/7 and brackets.
• Limits. Incorrect limits, reversed limits, not declaring limits at all, and errors pairing up the limits with the probability value. In another question, errors when limits were substituted.
• Not finding the domain first. One question needed the domain of g(x) to be found before integrating.
• Sign errors. Often from omitting the negative sign of the function.
• The equation of a line. Using a positive gradient rather than a negative one, using c = 0 when c = 1 was correct, or assuming y = x or a horizontal line.
• Mean and variance. Omitting to subtract the mean squared, or not squaring E(X) when subtracting. Another group found E(X²) or E(X) and did not go on. Some found E(X) = a³/6 but did not substitute a.
• Using symmetry. Many candidates did not mention symmetry, and thought the mean was 2 because the mean of the end points 1 and 3 was 2. Others only stated how to find E(X) by integration, without using the symmetry of the function.
• The median. Equating the integral to 0.5 is the standard method, and many did it well, obtaining a quadratic. A common error was to treat the median as the midpoint between the origin and a, and deduce a wrong value. Another was to think that q is twice the value of p, and integrate between −q and q.
• Rejecting a solution. Excluding a value such as k = 2 correctly but without justification. The reports say a valid reason is needed.
• Validity of a function. Very few candidates stated f(x) ≥ 0, which is also a requirement.
• Using geometry. For a semicircle, many used a set method of integration instead of the formula for the area of a semicircle equated to 1.
• Integration. Not attempting integration by parts, sign errors in it, and difficulty with e^(−λx). In another question, there were errors when integrating x²f(x).
• Notation. Once an expression has been integrated, the integral sign should no longer be used and should be replaced by brackets.
• Following instructions. Not using the answer to an earlier part when told to, and instead integrating between other limits.
• Not rigorous enough. In a question requiring a comparison and conclusion, appropriate comparisons or a valid conclusion were often missing.

What examiners advise

• Write down the integral and its limits before integrating, and state what it equals.
• Use symmetry to find the mean and median when the function allows it, and state that you are doing so.
• Check the function is valid: f(x) ≥ 0 and total probability 1.
• Give a reason whenever you reject a root.
• Use the answer to the previous part when the question says to.
• Show the full working for a given answer, with no missing factors or brackets.

Habits that help

• Sketch the pdf before you start. It shows you the limits, the symmetry and the sign of the gradient.
• For a straight-line pdf, find the gradient and intercept from two points and check it passes through both.
• After calculating Var(X), check that it is positive.

Questions students ask

How do I find an unknown constant in a pdf?

Write the integral with its limits and state clearly that it equals 1. Reports record students setting it equal to the wrong value, leaving out constants or factors, or using incorrect or reversed limits.

How do I find the median of a continuous random variable?

Equate the integral from the lower limit to m to 0.5, which often gives a quadratic. A common error was treating the median as the midpoint between the origin and a. Give a reason whenever you reject a root.

How do I find the mean and variance?

Integrate xf(x) for the mean and x²f(x) for E(X²), then subtract the mean squared. Students omitted to subtract or did not square E(X). Use symmetry where the function allows it, and say so.

What makes a valid pdf?

f(x) ≥ 0 for all x and total probability 1. Very few candidates stated f(x) ≥ 0.

Do I need to integrate if the pdf is a semicircle or straight line?

You can use geometry. For a semicircle, equate the area to 1 using the semicircle area formula. For a straight line, find the gradient and intercept from two points and check it passes through both.