Exercises: Random variable. Probability mass function, cumulative distribution function and parameters

A random number app generates integers from $1$ to $4$, all with equal probability. Let $X$ be the number generated.

1. a) Write down the probability function of $X$ in table form.
2. b) Calculate the mean $\mu$.
3. c) Calculate the standard deviation $\sigma$, rounding to two decimal places.

A discrete random variable $X$ takes the values $1$, $2$, $3$ and $4$ with this incomplete probability function:

$x_i$	$1$	$2$	$3$	$4$
$p_i$	$0.15$	$k$	$0.35$	$0.25$

1. a) Calculate the value of $k$.
2. b) Find $P[X \ge 3]$.
3. c) Find $P[X < 2]$.

State, giving reasons, which of the following tables can represent the probability function of a discrete random variable:

• Table 1: values $1$, $2$, $3$ with probabilities $0.3$; $0.4$; $0.2$.

• Table 2: values $1$, $2$, $3$, $4$ with probabilities $0.5$; $0.3$; $0.3$; $-0.1$.

• Table 3: values $1$, $2$, $3$, $4$ with probabilities $0.25$; $0.25$; $0.25$; $0.25$.

In each case, state the condition (or conditions) that are satisfied or violated.

Two coins are flipped and the number of heads obtained, $X$, is counted.

1. a) Write down the sample space of the experiment and deduce the probability function of $X$.
2. b) Calculate the mean $\mu$ and interpret it.
3. c) Calculate $P[X \ge 1]$.

The cumulative distribution function of a discrete random variable $X$ is given by:

$F(0) = 0.2 \qquad F(1) = 0.5 \qquad F(2) = 0.8 \qquad F(3) = 1$

1. a) Calculate $P[X \le 2]$.
2. b) Calculate $P[X = 2]$.
3. c) Calculate $P[X > 1]$.

In a school rapid chess championship in Cork, the score obtained per game is modelled by a discrete random variable $X$ with mean $\mu_X = 4$ points and standard deviation $\sigma_X = 1.2$ points. For the final qualifying stage, the organisation applies a correcting scale defined by the linear transformation $Y = 20 - 3X$.

Cian states in his homework submission: "The new mean is $\mu_Y = 20 - 3 \times 4 = 8$ points, and the new standard deviation is $\sigma_Y = -3 \times 1.2 = -3.6$ points; therefore, the scores are less dispersed because the sign of the scale has been inverted."

1. a) Identify and explain in detail the conceptual error made by Cian in the calculation of the standard deviation.
2. b) Correct the calculations by finding the correct values of $\mu_Y$ and $\sigma_Y$.
3. c) Justify in your own words why a standard deviation can never take a negative value, relying on its statistical definition as a measure of dispersion.

Based on simplified data from the Central Statistics Office (CSO) regarding household equipment in Ireland, the number of passenger cars $X$ owned by a household is modelled by the following probability function:

$P[X = 0] = 0.20$, $P[X = 1] = 0.55$, $P[X = 2] = 0.20$, and $P[X = 3] = 0.05$.

1. a) Before carrying out any numerical calculations, make a reasoned estimate: between which two integer values must the mean $\mu$ necessarily lie, and which of them do you think it will be closer to? Justify this by analysing where the highest probability is concentrated.
2. b) Calculate the mean $\mu$ and the standard deviation $\sigma$ exactly, and compare the value obtained for the mean with your previous estimate.
3. c) A neighbourhood spokesperson states: "Since the mean value is not an integer number of cars, the mean lacks physical meaning to describe a real household." Refute their statement by explaining precisely what mathematical expectation represents in the long run.

In a bag for a board game there are $20$ point cards: $6$ cards with a $1$, $4$ cards with a $2$ and $10$ cards with a $3$. A card is drawn at random and its score $X$ is noted.

1. a) Write down the probability function of $X$ in table form.
2. b) Calculate the mean $\mu$ and interpret it: what average score per card can be expected in the long run?
3. c) Calculate the standard deviation $\sigma$, rounding to two decimal places.

The figure shows the cumulative distribution function $F(x)$ of a discrete random variable $X$ that takes the values $0$, $1$, $2$ and $3$.

1. a) Deduce from the graph the probability function of $X$ (the values $p_i = P[X = x_i]$) and write it down in table form.
2. b) Calculate $P[1 \le X < 3]$ using the cumulative distribution function.
3. c) Find the mean of $X$.

A roulette wheel at a fair has $8$ equal sectors with the prizes shown in the figure: one sector with $\text{€}\ 10$, two with $\text{€}\ 5$, three with $\text{€}\ 2$ and two with no prize.

1. a) Write down the probability function of the variable $X$ = "prize obtained in a spin".
2. b) Calculate the average prize per spin, $\mu$.
3. c) Playing costs $\text{€}\ 3$ per spin. Define the variable $G$ = "player's net winnings" (prize minus amount paid), calculate its expected value and decide, giving reasons, whether it is advisable to play.

The PTA of a secondary school organises a charity raffle: it sells $200$ tickets at $\text{€}\ 2$ each, with one prize of $\text{€}\ 100$, four prizes of $\text{€}\ 25$ and ten prizes of $\text{€}\ 5$.

1. a) Write down the probability function of the variable $G$ = "net winnings from buying a ticket" (value of the prize minus the $\text{€}\ 2$ it costs).
2. b) Calculate the expected value of $G$.
3. c) Interpret the sign of the result: if the expected value is negative, where does that money go? Does that contradict the charitable purpose of the raffle?

A discrete random variable $X$ takes the values $1$, $2$, $3$ and $4$ with probabilities proportional to the value itself: $P[X = x] = k \times x$.

1. a) Calculate the value of the constant $k$.
2. b) Find the mean $\mu$.
3. c) Verify that the variance is exactly $1$ and deduce the standard deviation.

The waiting time $X$ (in minutes) for the school bus at a stop is uniformly distributed between $0$ and $10$ minutes, that is, its probability density function is constant: $f(x) = k$ for $x \in [0, 10]$ and $f(x) = 0$ elsewhere.

1. a) Calculate the value of $k$ knowing that the area under the probability density function must be $1$.
2. b) Find the probability of waiting between $2$ and $5$ minutes.
3. c) Find the probability of waiting more than $8$ minutes and explain why $P[X = 8] = 0$ even though waiting for exactly $8$ minutes is not impossible.

To fund their 6th Year school trip, the students design a charity roulette wheel. The random variable $X$ represents a person's net gain in euros after one spin, and it only takes the values $-3$ € (if they lose their bet), $+1$ € (minor prize), and $+k$ € (grand prize, with $k > 1$). It has been established that the probability of losing is $P[X = -3] = 0.40$ and that of winning one euro is $P[X = 1] = 0.45$.

1. a) Determine the probability $P[X = k]$ of winning the grand prize, justifying which property of probability distributions you use.
2. b) Set up the expected value equation and calculate the exact value of $k$ so that the game is fair ($E[X] = 0$).
3. c) If the organising team wants to guarantee an expected average collection for the trip of $0.45$ € for each spin sold (that is, that the player's average net gain is $E[X] = -0.45$ €), what integer value should be assigned to the prize $k$?

An express courier company operating in Cork analyses the number of failed visits $X$ before being able to deliver a parcel in person. The estimated probability model for each delivery is as follows:

$P[X = 0] = 0.60$; $P[X = 1] = 0.25$; $P[X = 2] = 0.10$; $P[X = 3] = 0.05$, with a fourth failed attempt never occurring.

1. a) Write the cumulative distribution function $F(x) = P[X \le x]$ for any real number $x \in \mathbb{R}$, clearly indicating the intervals and identifying the jump value that occurs at each possible value of the variable.
2. b) Each failed attempt entails an operating cost of $3$ € for fuel and the courier's time, plus a fixed handling cost of $2$ € per delivered parcel. Express the total cost per parcel $C$ as a linear transformation of $X$ ($C = aX + b$), and calculate its mean cost $E[C]$ and its standard deviation $\sigma_C$.
3. c) If the company delivers $1,000$ parcels in a day, how much total budget does it expect to allocate to these costs? Explain which probabilistic principle ensures that the actual average cost for that day will closely approximate $E[C]$.

The time (in minutes) it takes for a server to boot after a restart is a continuous random variable $X$ with probability density function $f(x) = k \times x$ for $x \in [0, 4]$ and $f(x) = 0$ elsewhere, represented in the figure.

1. a) Calculate the value of $k$ knowing that the total area under the probability density function must be $1$.
2. b) Find the probability that the server takes less than $2$ minutes to boot.
3. c) Calculate the median of $X$ (the value $m$ such that $P[X \le m] = \frac{1}{2}$) and the mean $\mu$. Which of the two is greater? What do you think is the reason for this?

Amina is studying the variable $X$ = "number of weekly breakdowns of a photocopier", with this probability distribution:

$x_i$	$0$	$1$	$2$	$3$
$p_i$	$0.1$	$0.3$	$0.4$	$0.2$

She has calculated the mean, $\mu = 1.7$, and then wrote:

«$\sigma^2 = \sum p_i \times x_i^2 - \mu = 3.7 - 1.7 = 2 \implies \sigma = \sqrt{2} \approx 1.41$»

1. a) Check her calculation of $\sum p_i \times x_i^2$ and confirm that it is correct.
2. b) Locate the error in the formula she applied for the variance and correct it.
3. c) Calculate the correct standard deviation.

At a fairground stall, a game is played with two cubic dice: you win $\text{€}\ 12$ if you roll doubles (both dice the same), $\text{€}\ 6$ if the sum is $7$, and nothing in any other case.

1. a) Calculate the probability of winning each prize and the probability of winning nothing. (Note that rolling doubles and having a sum of $7$ are mutually exclusive events; why?)
2. b) Find the expected value of the prize per game and deduce what the price of the stake should be for the game to be fair.
3. c) If the stall charges $\text{€}\ 4$ per game, who does the game favour? How much is a player who plays $100$ games expected to win or lose?

A discrete random variable $X$ takes the values $0$, $1$, $2$ and $4$, and its cumulative distribution function satisfies:

$F(0) = 0.1 \qquad F(1) = 0.5 \qquad F(2) = a \qquad F(4) = 1$

1. a) Knowing that the mean of the distribution is $\mu = 2.2$, set up the equation that allows you to find $a$ and calculate its value.
2. b) Write down the complete probability function of $X$.
3. c) Calculate the variance and standard deviation of $X$, rounding to two decimal places.

Let $X$ be a discrete random variable with mean $\mu$ and standard deviation $\sigma$, and let $Y = a \times X + b$ be a new variable obtained by applying a linear transformation to $X$.

1. a) Prove, from the definition of the mean, that $\mu_Y = a \times \mu + b$.
2. b) Prove that $\sigma_Y = |a| \times \sigma$. Why does the constant $b$ not affect the spread?
3. c) Application: in an exam, the marks have a mean of $6.2$ and a standard deviation of $1.4$. The teacher decides to transform each mark $X$ into $Y = 0.9 \times X + 1$. Calculate the new mean and the new standard deviation.

To analyse the variability of scores in a selection test, a discrete variable $X$ is defined that takes only four large values: $200$, $204$, $208$, and $212$, with associated probabilities $0.15$, $0.35$, $0.30$, and $0.20$, respectively. To calculate the standard deviation $\sigma_X$, two methods are discussed:

• Method A (Direct calculation): Calculate $\mu_X = \sum x_i p_i$, then $\sum x_i^2 p_i$ (handling values greater than $40,000$), and apply $\sigma_X = \sqrt{\sum x_i^2 p_i - \mu_X^2}$.

• Method B (Linear change of variable): Define a scaled variable $Y = \frac{X - 200}{4}$, calculate the mean $\mu_Y$ and the standard deviation $\sigma_Y$ of this simplified variable (whose values are only $0, 1, 2, 3$), and from them recover $\mu_X$ and $\sigma_X$.

1. a) Apply Method B: write the probability distribution of $Y$, calculate $\mu_Y$ and $\sigma_Y$, and determine $\mu_X$ and $\sigma_X$ from them using the properties of linear transformations.
2. b) Apply Method A to verify the result and compare both procedures: what risk of rounding error or loss of numerical precision does Method A present compared to Method B when working with calculators of limited precision?
3. c) Justify theoretically why subtracting $-200$ shifts the mean but does not alter dispersion at all, whereas dividing by $4$ scales both the mean and the standard deviation.

The cumulative distribution function of the daily number of technical incidents $X$ on a commuter train line is given by:

$$F(x) = \begin{cases} 0 & \text{if } x < 0 \\ 0.40 & \text{if } 0 \le x < 1 \\ 0.75 & \text{if } 1 \le x < 2 \\ 0.95 & \text{if } 2 \le x < 3 \\ 1 & \text{if } x \ge 3 \end{cases}$$

1. a) Reconstruct the probability function of $X$ (indicating the value of $P[X = x_i]$ for each possible value) from the jumps of $F(x)$, and check that it satisfies the two conditions necessary to be valid.
2. b) It is known that on a given day at least one incident occurred ($X \ge 1$). Calculate the conditional probability that exactly $2$ incidents were recorded on that day.
3. c) The maintenance company charges a daily fixed fee of $100$ € that covers up to one incident (if there are $0$ or $1$ incidents, no extra charge is paid); however, starting from the second incident (that is, for the second and for the third), each one generates an additional cost of $150$ €. Model the random variable $G$ (daily expense in euros), determine its probability function, calculate the expected average expense $E[G]$, and explain why $G$ cannot be expressed as a single global linear transformation of the form $aX + b$ for all values of $X$.