Mathematics 9709/52 — May/June 2023

Walkthrough

The key fact for any probability distribution is that the probabilities of all possible outcomes must add to $1$. Here the probability formula is $P(X=x)=k(x^2-1)$. Substitute the three values $-2$, $2$ and $3$ into the formula to get $3k$, $3k$ and $8k$. Add these and set the sum equal to $1$: $3k+3k+8k=1$, so $14k=1$ and $k=\frac{1}{14}$. Finally, replace $k$ in each expression to obtain the probabilities $\frac{3}{14}$, $\frac{3}{14}$ and $\frac{8}{14}$, which simplifies to $\frac{4}{7}$. The table must list the outcomes $-2$, $2$, $3$ with these probabilities.

Key Takeaways

A probability distribution table lists every possible value of the random variable with its probability. The probabilities must sum to $1$, which is the equation used to find an unknown constant in a probability formula.

Common Mistakes

• Forgetting that the probabilities must sum to $1$, so not forming $3k+3k+8k=1$.
• Substituting incorrectly, for example using $x^2+1$ instead of $x^2-1$.
• Not simplifying $\frac{8}{14}$ to $\frac{4}{7}$; this is not wrong, but simplified fractions are usually preferred.
• Adding extra outcomes to the table without giving probability $0$.

Things to Be Careful About

The mark scheme allows follow-through using a candidate's own $k$. If $k$ is not calculated, a table with probabilities $3k$, $3k$, $8k$ can still earn special credit. Also, the outcomes in the table must be exactly $-2$, $2$ and $3$.

Walkthrough

For expectation, use $E(X)=\sum xP(X=x)$. Multiply each value of $X$ by its probability from part (a) and add:

For variance, use $\mathrm{Var}(X)=E(X^2)-[E(X)]^2$. First find $E(X^2)=\sum x^2P(X=x)$:

Then subtract $[E(X)]^2=\left(\frac{12}{7}\right)^2=\frac{144}{49}$:

The final answers should be identified as expectation and variance.

Key Takeaways

Expectation is a weighted average using probabilities as weights. Variance measures spread and can be calculated as $E(X^2)-[E(X)]^2$, which is often easier than summing squared deviations directly.

Common Mistakes

• Using $E(X^2)$ as the variance without subtracting $[E(X)]^2$.
• Forgetting to square the negative value $-2$ when calculating $E(X^2)$; $(-2)^2=4$.
• Mixing up $E(X)$ and $E(X^2)$.
• Not identifying which answer is $E(X)$ and which is $\mathrm{Var}(X)$; the mark scheme requires them to be identified.

Things to Be Careful About

The mark scheme allows follow-through from part (a) using the candidate's table, provided the probabilities are valid. If the final answers are correct but not identified, the final accuracy mark may be lost. If the final accuracy mark is not earned, a special case mark is available for identified correct final answers.