Random variables

Introduce continuous random variables and probability density functions; calculate probabilities as areas under a pdf; find mean, median and mode of continuous distributions; apply the normal distribution including z-scores and the empirical rule.

Statistics and Probability · MAV-12-07

Sub-topics

Continuous random variables

Define a probability density function (pdf); calculate probabilities as definite integrals; find the mean, median and mode of a continuous distribution; work with uniform and simple polynomial pdfs.

The normal distribution

Explore the normal distribution as a continuous probability distribution; standardise values using z-scores; apply the empirical (68–95–99.7%) rule to find probabilities and proportions; compare values across different normal distributions.

Open Math

Random variables

Statistics and Probability · MAV-12-07

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Date:

Continuous random variables

Q1Straightforward

A continuous random variable

X

has probability density function

f(x) = kx

for

0 \le x \le 2

, and

f(x) = 0

otherwise.

Find the value of

k

so that

f

is a valid pdf.

Q2Straightforward

X

is uniformly distributed on the interval

[3, 9]

, so

f(x) = \dfrac{1}{6}

for

3 \le x \le 9

.

Find

P(4 \le X \le 7)

Q3Straightforward

X

has pdf

f(x) = 3x^2

for

0 \le x \le 1

(and

0

elsewhere).

Find

P(X \le 0.5)

Q4Straightforward

X

has pdf

f(x) = 3x^2

for

0 \le x \le 1

.

Find the mode of

X

(the value of

x

at which

f(x)

is largest).

Q5Moderate

X

has pdf

f(x) = \dfrac{x}{8}

for

0 \le x \le 4

(and

0

elsewhere).

Find

P(1 \le X \le 3)

Q6Moderate

X

is uniformly distributed on

[0, 4]

, with pdf

f(x) = \dfrac{1}{4}

for

0 \le x \le 4

.

Find the mean

E[X]

Q7Moderate

X

has pdf

f(x) = 3x^2

for

0 \le x \le 1

.

Find the mean

E[X]

. Express your answer as a decimal.

Q8Moderate

X

has pdf

f(x) = 2(1 - x)

for

0 \le x \le 1

.

Find the median of

X

, rounded to 4 decimal places.

(The median

m

satisfies

\displaystyle\int_0^m f(x)\,dx = 0.5

Q9Challenging

X

has pdf

f(x) = cx(4 - x)

for

0 \le x \le 4

, and

f(x) = 0

otherwise.

Find the value of

c

. Express your answer as a fraction in the form

p/q

Q10Challenging

X

has pdf

f(x) = \dfrac{3}{32}x(4 - x)

for

0 \le x \le 4

.

Find

E[X]

Q11Challenging

X

has pdf

f(x) = 6x(1 - x)

for

0 \le x \le 1

.

Find

P\!\left(X \le \dfrac{1}{3}\right)

. Express your answer as a fraction

p/q

in simplest form.

Q12Challenging

X

has pdf

f(x) = 6x(1 - x)

for

0 \le x \le 1

.

Find the mean

E[X]

The normal distribution

Q13Straightforward

X \sim N(40, 25)

, so

\mu = 40

and

\sigma = 5

.

Find the

z

-score corresponding to

x = 45

Q14Straightforward

X \sim N(70, 16)

, so

\mu = 70

and

\sigma = 4

.

A value in this distribution has

z

-score

z = -2

. Find the value of

x

Q15Straightforward

X \sim N(20, 9)

, so

\mu = 20

and

\sigma = 3

.

Using the empirical rule, find

P(17 \le X \le 23)

Q16Straightforward

The lifespans of a species of insect are normally distributed with mean 24 days and standard deviation 4 days. Using the empirical rule, what proportion have lifespans between 16 and 32 days?

Q17Moderate

Scores on an aptitude test are modelled by

X \sim N(100, 225)

, so

\mu = 100

and

\sigma = 15

.

Using the empirical rule, find

P(X > 115)

Q18Moderate

X \sim N(50, 4)

, so

\mu = 50

and

\sigma = 2

.

Using the empirical rule, find

P(50 \le X \le 54)

Q19Moderate

X

is normally distributed with mean

\mu = 80

. Using the empirical rule, 99.7% of values fall between 65 and 95. Find the standard deviation

\sigma

Q20Moderate

In a Mathematics exam, a student scored 78 in a class where

\mu = 70

and

\sigma = 10

. Find the student's

z

-score for Mathematics.

Q21Moderate

The same student also sat a Science exam, scoring 85 in a class where

\mu = 80

and

\sigma = 8

. Find the student's

z

-score for Science. Round to 3 decimal places if needed.

Q22Moderate

The heights of plants in a large population are normally distributed with mean 45 cm and standard deviation 6 cm. In a sample of 2000 plants, how many would you expect to have heights between 33 cm and 57 cm? (Use the empirical rule.)

Q23Challenging

X \sim N(200, 400)

, so

\mu = 200

and

\sigma = 20

.

Using the empirical rule, find

P(180 \le X \le 240)

Q24Challenging

A normally distributed random variable

X

has standard deviation

\sigma = 6

. A value of

x = 53

corresponds to a

z

-score of

z = 2

. Find the mean

\mu

Q25Challenging

X \sim N(150, 100)

, so

\mu = 150

and

\sigma = 10

.

Using the empirical rule, find

P(X < 130)

Worked solutions and answers at openmath.au/year-12/advanced/random-variables