Exercises: Discrete probability function and cumulative distribution function

Determine with reasons whether the following table can represent the probability function of a discrete random variable $X$. To do this, check whether the fundamental properties of probabilities are satisfied:

$x_i$	$1$	$2$	$3$	$4$
$P(X = x_i)$	$0,25$	$0,40$	$0,15$	$0,20$

A streaming platform analyses the number of series a user watches per week. The discrete random variable $X$ is defined as the "number of completed series". Its probability function is given by the following table:

$x_i$	$0$	$1$	$2$	$3$
$P(X = x_i)$	$0,1$	$k$	$0,3$	$0,2$

Find the value of the constant $k$ for the distribution to be well-defined, and calculate the probability that a user watches at least 2 series a week, that is, $P(X \ge 2)$.

Two fair coins are flipped. We define the random variable $X$ as the "number of heads obtained".

1. a) Write the sample space of the experiment.
2. b) Construct the table of the probability function of $X$.
3. c) Represent the distribution graphically using a bar chart.

You can do it on paper and send a photo with your answer.

Given the following probability function of a discrete random variable $X$, calculate the probabilities requested below:

$x_i$	$10$	$20$	$30$	$40$	$50$
$p_i$	$0,15$	$0,25$	$0,35$	$0,15$	$0,10$

1. a) $P(X < 30)$
2. b) $P(X \le 30)$
3. c) $P(20 < X \le 40)$

An urn contains 5 numbered balls: three of them with the number $1$ and two of them with the number $2$. A ball is drawn at random and its value is noted.

1. a) Define the random variable $X$ and its range (possible values).
2. b) Construct the table of the probability function.
3. c) What is the probability that the number obtained is greater than $1$?

You can do it on paper and send a photo with your answer.

Let $X$ be a discrete random variable whose probability function is given by:

$P(X=x) = \frac{x}{10}$ for $x \in \{1, 2, 3, 4\}$

1. a) Write the table of values for the probability function.
2. b) Check that the sum of all probabilities is equal to $1$.
3. c) Calculate $P(X \ge 3)$.

A survey was conducted in a secondary school regarding the number of siblings of 1st Bachillerato students. From the data, the variable $X$ = "number of siblings" was defined. The relative frequency table (estimated probabilities) is as follows:

Siblings ($x_i$)	$0$	$1$	$2$	$3$
$P(X = x_i)$	$0,12$	$0,54$	$0,24$	$0,10$

Represent this information using a bar chart and calculate the probability that a student chosen at random has fewer than 2 siblings.

You can do it on paper and send a photo with your answer.

Explain why the following function cannot be the probability function of a discrete random variable $X$:

$P(X=x_i) = \{0,2;\ 0,3;\ -0,1;\ 0,6\}$

The figure shows these values assigned to $x_1, x_2, x_3$ and $x_4$.

Justify your answer by mentioning which property of probability is being violated in this case.

A telecommunications company analyzes the number $X$ of formal complaints filed by a customer over the course of a year. The cumulative distribution function of $X$ is given by:

$$F(x) = \begin{cases} 0 & \text{if } x < 0 \\ 0,55 & \text{if } 0 \leq x < 1 \\ 0,75 & \text{if } 1 \leq x < 2 \\ 0,90 & \text{if } 2 \leq x < 3 \\ 1 & \text{if } x \geq 3 \end{cases}$$

Marcos, a trainee analyst, must calculate two probabilities and submits the following written answers:

1) "The probability that a customer files exactly $2$ complaints is $P(X = 2) = F(2) = 0,90$".

2) "The probability that a customer files at least one complaint is $P(X \geq 1) = 1 - F(1) = 1 - 0,75 = 0,25$".

Identify, with reasoned justification, the two conceptual errors made by Marcos, explain what the values he obtained actually represent, and find the correct values of $P(X = 2)$ and $P(X \geq 1)$ by applying the properties of the distribution function.

An artisanal bakery bakes a maximum of $3$ signature cakes every morning. The number $X$ of cakes sold on a weekday follows a probability distribution given by the following table, where $k$ is a positive real constant:

$x_i$	$0$	$1$	$2$	$3$
$P(X=x_i)$	$0,15$	$0,25$	$3k$	$k$

1. a) Before calculating the value of $k$, make a reasoned estimate: do you think that the probability of selling at most one cake, $P(X \leq 1)$, will be greater than, less than, or equal to that of selling more than one cake, $P(X > 1)$? Justify your conjecture using the fundamental properties of the addition of probabilities.
2. b) Algebraically determine the exact value of the constant $k$ and write explicitly the cumulative distribution function $F(x)$ for the entire real line.
3. c) Calculate the exact value of $P(X \leq 1)$ and $P(X > 1)$. Check whether your initial estimate in section a) was correct and assess the consistency of the result.

In a workshop specialising in mobile device repair, a study was carried out on the number of devices a technician can fully repair in an eight-hour working day. Let $X$ be the random variable expressing this number. The probability function is given by the following table, although one value needs to be completed:

$x_i$	$0$	$1$	$2$	$3$	$4$
$P(X=x_i)$	$0,15$	$0,25$	$0,35$	$k$	$0,05$

1. a) Calculate the value of $k$ so that the table represents a valid probability distribution.
2. b) What is the probability that the technician repairs at least two devices in a day?
3. c) If the workshop receives a productivity bonus on days when more than 2 devices are repaired, what is the probability of receiving this bonus?

A youth association is organising a raffle to fund an end-of-year trip. Tickets are sold for $2$ euros. The prize is an Amazon gift card valued at $50$ euros. A total of $200$ tickets have been issued. Let $X$ be the random variable that defines the net gain of a student who buys a single ticket (taking into account its cost).

1. a) Define the sample space of the possible values of $X$.
2. b) Determine the probability function of $X$.
3. c) Calculate the expected value $E[X]$ and interpret the result obtained. Is it a fair game for the buyer?

A biased six-sided die is rolled, such that the probability of getting an even number is twice that of getting an odd number. If all even numbers are equally likely, and the same applies to odd numbers:

1. a) Determine the probability function of the random variable $X$ = "outcome obtained when rolling the die".
2. b) Calculate the probability of getting a value less than or equal to $3$.
3. c) Calculate the mean or expected value of this distribution.

A courier company analyses the number of damaged packages arriving in each delivery van. After a long observation period, they have defined the following probability function for the variable $X$ = "number of damaged packages per journey":

$P(X=x) = \frac{x+1}{15}$ for $x \in \{0, 1, 2, 3, 4\}$

1. a) Check that this is indeed a probability function.
2. b) Calculate the probability that at most $2$ damaged packages arrive on a journey.
3. c) Find the mean and standard deviation of the distribution.

Let $X$ be a discrete random variable whose cumulative distribution function $F(x)$ is shown below:

$$F(x) = \begin{cases} 0 & \text{if } x < 1 \\ 0,2 & \text{if } 1 \le x < 2 \\ 0,5 & \text{if } 2 \le x < 3 \\ 0,9 & \text{if } 3 \le x < 4 \\ 1 & \text{if } x \ge 4 \end{cases}$$

1. a) From $F(x)$, obtain the probability function $P(X=x_i)$ for each value of the variable.
2. b) Calculate the probability $P(2 \le X < 4)$.
3. c) Represent the probability function graphically using a bar chart.

You can do it on paper and send a photo with your answer.

In a language app, users can earn between $0$ and $3$ daily badges according to their consistency. It is known that the probability of earning $x$ badges is proportional to the value of $x+1$ for the possible values of the variable.

1. a) Write the probability distribution table, determining the required constant of proportionality.
2. b) What is the average number of badges a user earns per day?
3. c) Calculate the variance of the distribution.

A restaurant analyses the number of bookings for groups of more than 10 people it receives on Saturday nights. The variable $X$ follows the probability distribution:

$x_i$	$0$	$1$	$2$	$3$
$P(X=x_i)$	$0,4$	$0,3$	$0,2$	$0,1$

1. a) If the restaurant makes a profit of $50$ euros for each group that attends, but incurs fixed preparation costs of $30$ euros regardless of how many groups go, define a new variable $Y$ representing the net profit for Saturday.
2. b) Calculate the expected value of $Y$.
3. c) Calculate the standard deviation of $X$.

There is an urn with $3$ white balls and $2$ red balls, as shown in the figure. Two balls are drawn successively without replacement. Let $X$ be the random variable that counts the number of white balls drawn.

1. a) Describe the sample space of the experiment and the values that $X$ can take.

1. b) Calculate the probability function of $X$.

1. c) Determine the probability that at least one of the drawn balls is red.

Based on microdata adapted from the Essential Population and Housing Characteristics Survey of the INE, the number $X$ of passenger cars owned per household in a Spanish municipality is studied. It is known that:

• $20\,\%$ of households do not own any passenger car.

• Half of the households ($50\,\%$) have exactly one passenger car.

• The probability of owning $2$ passenger cars is exactly three times that of owning $3$ passenger cars.

• No household in the sample owns $4$ or more passenger cars.

1. a) Formally define the random variable $X$, state its support (values it can take), and determine its complete probability mass function in the form of a table.
2. b) Obtain the analytical piecewise expression of the cumulative distribution function $F(x)$ for any real number $x$.
3. c) Using exclusively the distribution function $F(x)$, calculate the probability that a randomly selected household has at least one passenger car but no more than two ($P(1 \leq X \leq 2)$) and the probability that it has strictly fewer than $2$ passenger cars. Interpret the results obtained in the socio-economic context of the municipality.

A discrete random variable $X$ takes exclusively the integer values $\{1, 2, 3, 4\}$. The individual probabilities associated with each value are unknown, but the following information about its cumulative behavior is available:

• The probability that the variable takes the minimum value is $P(X = 1) = 0,20$.

• The value of the distribution function at $x = 3$ is exactly $F(3) = 0,80$.

• The jump experienced by the distribution function $F(x)$ at the point $x = 2$ is exactly half the jump it experiences at the point $x = 3$.

1. a) Reasonably deduce the complete probability mass function of the variable $X$, indicating the value of $P(X = x_i)$ for each $x_i \in \{1, 2, 3, 4\}$.
2. b) Write the distribution function $F(x)$ as a piecewise-defined function on the entire real line $\mathbb{R}$.
3. c) From the obtained distribution function, calculate with justification $P(1 < X \leq 3)$, $P(X > 2)$, and $P(X \leq 2,5)$.

Let $X$ be a discrete random variable whose probability function is defined in the following table, where $a$ and $b$ are two unknown real numbers:

$x_i$	$1$	$2$	$3$	$4$
$P(X=x_i)$	$0,1$	$a$	$b$	$0,2$

Given that the expected value of the distribution is $E[X] = 2,85$, determine the values of $a$ and $b$. Subsequently, calculate the standard deviation $\sigma$ of the variable and indicate what percentage of the total probability lies in the interval $[E[X] - \sigma, E[X] + \sigma]$.

A logistics company organizes a darts tournament for its employees. The probability of hitting the red zone is $k$, the blue zone is $2k$ and the white zone is $3k$. If a player throws two darts independently, we define the random variable $X$ as "the number of times the dart hits the red zone".

1. Find the value of $k$ for the probability distribution to be valid (assume the dart always hits one of the three zones).

2. Obtain the probability function of $X$.

3. Calculate the expected number of hits in the red zone if the employee performs $50$ series of two throws each.

The cumulative distribution function $F(x)$ of a discrete random variable $X$ is given by a step graph with the following intervals: $F(x) = 0$ for $x < 0$, $F(x) = 0{,}2$ for $0 \le x < 1$, $F(x) = 0{,}5$ for $1 \le x < 2$, $F(x) = 0{,}8$ for $2 \le x < 3$, and $F(x) = 1$ for $x \ge 3$.

From the graph, construct the probability mass function table $P(X=x_i)$ and calculate the variance $V[X]$. Explain with reasoning the meaning of $P(0{,}5 < X \le 2)$.

You can work it out on paper and submit a photo of your answer.

A social media content creator analyses the daily number of "unsubscriptions" on their channel. They have found that the probability function of the variable $X$ (number of unsubscriptions) corresponds to the formula $P(X = x) = \frac{k \cdot x}{2^x}$ for $x \in \{1, 2, 3\}$.

1. Find the exact value of the constant $k$.

2. Calculate the probability that on a given day there are at least 2 unsubscriptions.

3. The creator estimates that each unsubscription costs them a loss of $5$ euros in advertising revenue, but gaining a new subscriber earns them $8$ euros. If they gain exactly $3$ new subscribers each day, calculate the expected average profit per day.

In an urn there are $10$ balls numbered from $1$ to $10$. Two balls are drawn simultaneously (without replacement). Let $X$ be the random variable that assigns to each draw the absolute value of the difference between the numbers obtained.

1. State the sample space of the experiment and the values that the variable $X$ can take.

2. What is the probability that the difference is maximum?

3. Without constructing the complete table, reason whether $X=1$ or $X=9$ is more likely. Calculate both probabilities.

Two players, Alba and Bruno, roll a six-sided die. If a prime number comes up, Alba wins as many euros as indicated on the die. If a composite number comes up, Bruno wins twice what the die indicates in euros. If the number $1$ comes up, nobody wins anything.

1. Define the random variable $G$ as Alba's net gain in a roll (considering that what Bruno wins is a loss for Alba). Write its probability function.

2. Calculate the mathematical expectation of $G$. Whom does the game favour in the long run?

3. What amount should Alba pay Bruno each time a composite number comes up so that the game is fair ($E[G]=0$)?

Consider a discrete random variable $X$ with the following probability function:

$P(X=0) = p$

$P(X=1) = q$

$P(X=2) = 0,2$

If we know that the expected value is $E[X] = 0,8$ and the variance is $V[X] = 0,56$, set up a system of equations and calculate the values of $p$ and $q$. Verify that the sum of the probabilities is equal to 1.

A competitive exam consists of $4$ topics chosen at random from a total of $10$. A candidate has prepared only $6$ topics. We define the random variable $X$ as "the number of topics the candidate knows out of the $4$ chosen".

1. Calculate the probability that the candidate does not know any of the chosen topics.

2. Determine the probability that they know exactly $2$ topics.

3. Find the expected value of the variable $X$ and interpret the result in the context of the exam. Is it likely that the candidate passes if they need to know at least $3$ topics?

The number of connection micro-outages experienced by a banking server during a peak hour is a discrete random variable $X \in \{0, 1, 2, 3, 4, 5\}$ whose cumulative distribution function is given by:

$$F(x) = \begin{cases} 0 & \text{if } x < 0 \\ 0,10 & \text{if } 0 \leq x < 1 \\ 0,30 & \text{if } 1 \leq x < 2 \\ 0,65 & \text{if } 2 \leq x < 3 \\ 0,85 & \text{if } 3 \leq x < 4 \\ 0,95 & \text{if } 4 \leq x < 5 \\ 1 & \text{if } x \geq 5 \end{cases}$$

It is desired to calculate the probability that more than one micro-outage but fewer than four are recorded, that is, $P(1 < X < 4)$, and the probability of recording at least two micro-outages, $P(X \geq 2)$. To this end, two strategies are proposed:

• Method 1 (Pointwise reconstruction): First obtain the complete probability mass function by calculating each jump $P(X = x_i) = F(x_i) - F(x_{i-1})$ and identify the values of the support that satisfy the condition to sum their individual probabilities.

• Method 2 (Direct cumulative operations): Express the required probabilities directly as differences of values of $F(x)$, appropriately transforming the endpoints according to the strict or non-strict inequalities.

1. a) Apply both methods in detail to calculate $P(1 < X < 4)$ and $P(X \geq 2)$, verifying that they lead to exactly the same numerical result.
2. b) Critically compare both methods: analyze which of the two is more efficient if the variable took integer values from $0$ to $100$ and explain what systematic conceptual error would be made if the formula $P(1 < X < 4) = F(4) - F(1)$, typical of continuous variables, were applied directly.

A sustainable urban transport company in Madrid monitors the daily number $X$ of electric vans requiring an unscheduled en-route recharge ($X \in \{0, 1, 2, 3, 4\}$). The logistics department proposes the following model for the cumulative distribution function of $X$, dependent on two unknown parameters $\alpha$ and $\beta$:

$$F(x) = \begin{cases} 0 & \text{if } x < 0 \\ 0,40 & \text{if } 0 \leq x < 1 \\ \alpha & \text{if } 1 \leq x < 2 \\ \alpha + \beta & \text{if } 2 \leq x < 3 \\ 0,90 & \text{if } 3 \leq x < 4 \\ 1 & \text{if } x \geq 4 \end{cases}$$

Historical records provide two operational conditions:

1) The event "requiring at most one unscheduled recharge" is $50\,\%$ more likely than the event "not requiring any unscheduled recharge".

2) The probability that exactly two unscheduled recharges occur is double the probability that exactly three occur.

1. a) Model both conditions by setting up the corresponding equations from the jumps of $F(x)$ and determine with justification the numerical values of $\alpha$ and $\beta$.
2. b) Formally justify that the obtained function satisfies all the necessary conditions to be a legitimate distribution function of a discrete variable (check the value of each point probability $p_i$, their total sum, and the monotonicity of $F$).
3. c) Each unscheduled recharge generates an operating cost of $80$ € if the daily total does not exceed two recharges. If more than two recharges occur, the overload of the service forces the hiring of emergency external assistance with a fixed flat rate of $400$ € for the entire day. If there are no recharges, the cost is $0$ €. Reasonably calculate the probability that the daily operating cost exceeds $100$ €.