Discrete random variables | CAIE 9709 Probability & Statistics 1 Notes

Probability & Statistics 1

Mathematics9709

Outcome first, number second

Toss two coins and let $X$ be the number of heads. HH, HT, TH and TT are outcomes. The corresponding values of $X$ are 2, 1, 1 and 0.

Definition

1 mark

What is meant by a discrete random variable?

Model answer: A variable whose value is determined by chance and can take only specific, separate numerical values.

Key idea

The outcome and the value are different. HT and TH are different outcomes, but both give

X=1

List values the situation allows

Write exactly what the variable counts or measures.
Find the smallest possible value.
Find the largest possible value.
Check every value between those limits is achievable.

Worked example

A supply limit changes the maximum

Three people are selected from 4 boys and 2 girls. Let $G$ be the number of girls selected.

The minimum is 0: selecting three boys is possible.
The maximum is 2: only two girls are available.

Answer

G\in\{0,1,2\}

Common mistake

Do not automatically write 0 up to the sample size. Limited supplies can force a positive minimum or a smaller maximum.

Read X, x and P(X = x) differently

$X$ names the random variable.
$x$ stands for one possible numerical value.
$P(X=x)$ is the probability that $X$ takes that value.
$X\in\{0,1,2\}$ means that 0, 1 and 2 are the only possible values.

A count is usually discrete because there are gaps between possible values. A measured time can take any value in an interval, so it is continuous instead.

Your turnIndependent check [3 marks]

Two counters are selected from 3 red and 2 blue counters. Let $R$ be the number of red counters selected. State the possible values and find $P(R=2)$ .

Show worked answer

Zero, one or two red counters can be selected. For two reds, choose 2 of the 3 red counters from all selections of 2 from 5.

Answer

R\in\{0,1,2\}

P(R=2)=\frac{{}^3C_2}{{}^5C_2}

P(R=2)=\frac3{10}

Outcome first, number second

Toss two coins and let $X$ be the number of heads. HH, HT, TH and TT are outcomes. The corresponding values of $X$ are 2, 1, 1 and 0.

Definition

1 mark

What is meant by a discrete random variable?

Model answer: A variable whose value is determined by chance and can take only specific, separate numerical values.

Key idea

The outcome and the value are different. HT and TH are different outcomes, but both give

X=1

List values the situation allows

Write exactly what the variable counts or measures.
Find the smallest possible value.
Find the largest possible value.
Check every value between those limits is achievable.

Worked example

A supply limit changes the maximum

Three people are selected from 4 boys and 2 girls. Let $G$ be the number of girls selected.

The minimum is 0: selecting three boys is possible.
The maximum is 2: only two girls are available.

Answer

G\in\{0,1,2\}

Common mistake

Do not automatically write 0 up to the sample size. Limited supplies can force a positive minimum or a smaller maximum.

Read X, x and P(X = x) differently

$X$ names the random variable.
$x$ stands for one possible numerical value.
$P(X=x)$ is the probability that $X$ takes that value.
$X\in\{0,1,2\}$ means that 0, 1 and 2 are the only possible values.

A count is usually discrete because there are gaps between possible values. A measured time can take any value in an interval, so it is continuous instead.

Your turnIndependent check [3 marks]

Two counters are selected from 3 red and 2 blue counters. Let $R$ be the number of red counters selected. State the possible values and find $P(R=2)$ .

Show worked answer

Zero, one or two red counters can be selected. For two reds, choose 2 of the 3 red counters from all selections of 2 from 5.

Answer

R\in\{0,1,2\}

P(R=2)=\frac{{}^3C_2}{{}^5C_2}

P(R=2)=\frac3{10}