Random variables + effects linear transformations (mean, variance): notes and practice questions

The continuous random variable $X$ has probability density function

$f(x) = \begin{cases} \frac{k}{1+4x^2}, & 0 \le x \le \frac{1}{2} \\ 0, & \text{otherwise.} \end{cases}$

(a) Find the value of $k$.

(b) Find $E(X)$.

A new automated manufacturing process produces components. The time, in minutes, taken for a critical assembly step is modelled by a continuous random variable $X$, with a probability density function defined by

$f(x) = \begin{cases} \frac{2}{\pi \sqrt{9-x^2}} & 0 \leq x \leq 3 \\ 0, & \text{otherwise.} \end{cases}$

Find the exact value of $E(X)$.

Find $P(X < 1.5)$.

The assembly step is considered "efficient" if it takes less than 1.5 minutes. Each assembly step is independent. Determine the least number of assembly steps required to be 99% sure of at least one efficient step.

Ten assembly steps were conducted.

Find the probability that exactly three steps were efficient.

Write down the number of ways these three efficient steps could have occurred consecutively in a batch of 10.

Now consider a batch of $n$ assembly steps where it is given that exactly three efficient steps have occurred.

Write down an expression for the number of ways these three efficient steps could have occurred consecutively.

Find the greatest value of $n$ such that the probability of three consecutive efficient steps is more than 0.05, given that exactly three efficient steps have occurred in the batch.

A continuous random variable $Y$ has probability density function $g$ defined by

$g(y)=\begin{cases} \frac{1}{4c}, & -c \le y \le 3c \\ 0, & \text{otherwise} \end{cases}$

where $c$ is a positive real number.

(a) State $E(Y)$ in terms of $c$.

(b) Use integration to find $Var(Y)$ in terms of $c$.

A marine biologist is studying a species of sea turtle. She collects data on the carapace length, $L$ cm, and mass, $M$ kg, for 30 turtles. The Pearson's product-moment correlation coefficient for this data is found to be $r = -0.92$. The equation of the regression line of $M$ on $L$ is $M = -2.5L + 150$.

The biologist discovers her measuring tape was misaligned, and all length measurements are 2 cm too short. Her weighing scale was also faulty, showing a mass 1.5 kg less than the true mass for each turtle. The data is corrected for these errors.

(i) State the new value of the Pearson's product-moment correlation coefficient, $r$.

(ii) State the new value for the gradient of the regression line of $M$ on $L$.

(iii) Briefly justify your answers to part (a)(i) and (a)(ii).

The biologist decides to present her findings to an international conference and converts her original measurements to different units. She converts the original length measurements from cm to mm, and the original mass measurements from kg to g.

(i) State the new value of $r$.

(ii) Find the new value for the gradient of the regression line of mass on length.

(iii) Briefly justify your answer for the new gradient.

For a different analysis, the biologist defines a "size index", $S$, as $S = 200 - L$. She investigates the relationship between the size index $S$ and the original mass $M$ in kg.

(i) Find the value of $r$ for the correlation between $S$ and $M$.

(ii) Find the gradient of the regression line of $M$ on $S$.

(iii) Describe the linear correlation between the size index $S$ and the mass $M$.

A biologist records the number of eggs in the nests of a certain species of bird. The results for 25 nests are shown in the following frequency table.

It is given that the mean number of eggs per nest is 2.6.

(a) Find the value of p and the value of q.

The biologist enters the data into a competition. The competition score is calculated using the formula $S = 10x - 5$, where $x$ is the number of eggs in a nest.

(b) Find the mean competition score.

A continuous random variable $X$ has probability density function

$f(x) = \begin{cases} kx e^{-x} & x \ge 0 \\ 0 & \text{otherwise} \end{cases}$

Show that $k = 1$.

Find the variance of $X$.

A new coffee shop records the number of customers, $c$, per hour during its 120 opening hours in a week. The number of customers per hour is shown in the following frequency table.

(a) Find the value of $k$.

(b) Write down the modal class.

The following cumulative frequency diagram also displays these data.

(c) Use the cumulative frequency curve to estimate the median number of customers per hour.

(d) The coffee shop is considered 'busy' when there are more than 35 customers. Use the cumulative frequency curve to estimate the number of hours the coffee shop was busy.

The coffee shop manager wants to survey customers about their experience.

(e) State one disadvantage of surveying only the customers who arrive between 8 am and 9 am on a Monday.

(f) Describe how the manager could use systematic sampling to survey customers throughout a single day.

The total number of customers for the week was 2900. The following box and whisker diagram displays the amount of money, in USD, spent by customers during their visit.

(g) Estimate the number of customers who spent between $4.50 and $16.

(h) The top 25% of customers spent more than $d$ USD. Find the value of $d$.

The following week, a new promotion is introduced, which is expected to attract an additional 3 customers per hour.

(i) Calculate the new mean number of customers per hour.

(j) State, with a reason, the effect this increase would have on the range of the number of customers per hour.

This question asks you to investigate the motion of a buoy bobbing up and down in the water.

A buoy bobs up and down in the water.

A fixed origin $\text{O}$ is the equilibrium position of the buoy (the water level).

The buoy's displacement, $y$ metres, from $\text{O}$ at time $t$ seconds is given by

$y = 6\sin\left(2t + \frac{\pi}{6}\right), \text{ for } 0 \le t \le \pi.$

Determine

the amplitude of the buoy's motion;

the buoy's initial displacement from $\text{O}$;

the value of $t$ when the buoy first passes through $\text{O}$.

Now consider the general case of a buoy bobbing up and down.

The buoy's acceleration is always directed towards a fixed origin $\text{O}$ at its equilibrium position.

The buoy's acceleration, $a$, at a displacement, $y$, from $\text{O}$ satisfies the differential equation

$a = -\omega^2 y, \text{ where } \omega > 0.$

The buoy's displacement, $y$, from $\text{O}$ at time $t$ is given by

$y = H\sin(\omega t + c), \text{ where } t \ge 0,\ H, \omega > 0 \text{ and } -\pi \le c \le \pi.$

By finding expressions for $\frac{\mathrm{d}y}{\mathrm{d}t}$ and $\frac{\mathrm{d}^2y}{\mathrm{d}t^2}$, verify that $y = H\sin(\omega t + c)$ satisfies the differential equation $a = -\omega^2 y$.

Use the chain rule to show that $a = v\frac{\mathrm{d}v}{\mathrm{d}y}$, where $v$ is velocity.

By solving the differential equation, $v\frac{\mathrm{d}v}{\mathrm{d}y} = -\omega^2 y$, show that $v^2 = \omega^2(H^2 - y^2)$.

Hence, or otherwise, find the buoy's maximum speed.

The continuous random variable $Y$ denotes the buoy's displacement, $y$, from $\text{O}$ at time $t$.

The probability density function $f$ of $Y$ is defined by

$f(y) = \begin{cases} \frac{1}{\pi\sqrt{H^2 - y^2}}, & -H < y < H \\ 0, & \text{otherwise.} \end{cases}$

Show that $\mathrm{P}\left(0 \le Y \le \frac{H\sqrt{3}}{2}\right) = \frac{1}{3}$.

For $-H < y < H$, the function $f(y)$ can be expressed in the form $\frac{m}{|v(y)|}$, where $m > 0$ and $v(y)$ is the buoy's velocity at a displacement, $y$, from $\text{O}$.

Find the value of $m$.

Determine $\mathrm{E}(Y)$, justifying your answer.

Interpret the result found in part (f)(i) in the context of the buoy's motion.

A spinner has four sectors. When the spinner is spun, the score, Y, is a discrete random variable. The probability distribution of Y is given in the following table:

(a) Find the range of possible values of k.

(b) In the case where $k = 0.1$, determine $\operatorname{Var}(3Y+1)$.

The continuous random variable $Y$ has the following probability density function:

$g(y) = \begin{cases} \frac{a}{\sqrt{4 - y^2}}, & 0 \leq y \leq 1 \\ 0, & \text{otherwise} \end{cases}$

(a) Determine the value of $a$.

$b$ Find the expected value $E(Y)$

A manufacturing process produces components whose defect rate, $X$, is a continuous random variable. The probability density function for the defect rate $X$ is modelled by

$f_n(x) = \begin{cases} (n+1)x^n, & 0 \le x \le 1 \\ 0, & \text{otherwise} \end{cases}$

where $n \in \mathbb{R}, n \ge 0$ is a parameter reflecting the stringency of the quality control.

(a) Show that the expected defect rate, $E(X)$, is given by $E(X) = \frac{n+1}{n+2}$.

(b) Show that the variance of the defect rate, $Var(X)$, is given by $Var(X) = \frac{n+1}{(n+2)^2(n+3)}$.

A quality control manager inspects a batch of 40 components for defects. The number of defective components, $X$, follows a binomial distribution, $X \sim B(40, p)$. The variance of the number of defective components is known to be 8.4.

(a) Find the possible values of $p$.

The cost of repairing the batch of components, $Y$, is given by the formula $Y = 7 - 4X$ (in dollars).

(b) Find Var($Y$).

A set of daily temperature readings for a city, recorded in degrees Celsius ($C$), has a mean of $20$ and a standard deviation of $3$. These temperatures are converted to degrees Fahrenheit ($F$) using the formula $F = \frac{9}{5}C + 32$.

(a) Find the mean of the temperatures in degrees Fahrenheit.

(b) Find the standard deviation and the variance of the temperatures in degrees Fahrenheit.

A research team is studying the relationship between the average daily temperature ($x$ in $^\circ C$) during a growing season and the average height of a specific plant species ($y$ in cm) at harvest. They collected data over seven growing seasons.

(a) Write down the equation of the $y$ on $x$ regression line for this data, giving your coefficients to three significant figures.

(b) Estimate the average plant height if the average daily temperature during the growing season was $25^\circ C$.

Give your answer to one decimal place.

(c) The average daily temperatures were converted from Celsius to Fahrenheit using the formula $F = \frac{9}{5}C + 32$.

For each of the following quantities, state whether it would change or remain the same:

(i) the mean of the average daily temperatures

(ii) the standard deviation of the average daily temperatures

(iii) the correlation coefficient, $r$

A new model of industrial machine has an operating lifetime $X$ (in years) which is modelled by a continuous random variable with the following probability density function:

$$f(x) = \begin{cases} \frac{1}{5} & 0 \le x < 2 \\ -\frac{1}{40}x + \frac{1}{4} & 2 \le x \le 6 \\ 0 & \text{otherwise.} \end{cases}$$

(a) Calculate the expected operating lifetime of the machine, $E(X)$.

(b) The manufacturer offers an extended warranty for $k$ years. If the expected value of $(k - 3X)$ is $0$, determine the value of $k$.

(c) Find the median operating lifetime of the machine.

A farmer has $n$ plots of land. The total yield of wheat from all plots is 7200 kg and the mean yield per plot is 120 kg.

(a) Find the number of plots, $n$.

The standard deviation of the yield per plot is 15 kg. The farmer decides to report the yield in a different unit, where all yield values are multiplied by 2.

(b) (i) Write down the value of the new mean yield.

(ii) Find the value of the new variance.

A spinner used in a children's board game can land on one of six sectors, with scores 0, 1, 2, 3, 4, and 5. Let S be the discrete random variable representing the score from a single spin. The probability distribution of S is given in the table below.

(a) Calculate the value of $\text{E}(S^2)$.

(b) Find $\text{Var}(S)$.

A continuous random variable $T$ models the time, in hours, a student spends on a specific online learning module. The probability density function is given by

$f(t)=\begin{cases} 0.5t & 0 \le t \le 2 \\ 0 & \text{otherwise} \end{cases}$

(a) Determine the median time, $m$, spent on the module.

(b) Given that $P(|T-m| \le a) = 0.3$, determine the value of $a$.

A spinner is divided into eight equally sized sectors. The sectors are numbered 1, 2, 2, 3, 3, 3, 4, 4. Let $X$ be the discrete random variable representing the number obtained when the spinner is spun once.

(a) Complete the probability distribution table for $X$.

(b) Find the expected value of $X$.