Probability Function and Distribution Function of a Discrete Variable

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:

$x_i$	$1$	$2$	$3$	$4$
$P(X = x_i)$	$0.25$	$0.40$	$0.15$	$0.20$

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.

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 $P(X \ge 3)$.

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) $P(20 < X \le 40)$

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.

In a workshop specializing in mobile device repair, a study was carried out on the number of devices a technician can fully repair in an eight-hour workday. 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?

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 graph.

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.

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?

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.

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 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.

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.

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 youth association is organizing a raffle to fund an end-of-year trip. Tickets are sold for $40$ pesos. The prize is an Amazon gift card valued at $1\,000$ pesos. 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 restaurant analyzes the number of reservations 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 $1\,000$ pesos for each group that attends, but incurs fixed preparation costs of $600$ pesos 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 $X$.

Two players, Alba and Bruno, roll a six-sided die. If a prime number comes up, Alba wins as many pesos as indicated by the die. If a composite number comes up, Bruno wins double the amount indicated by the die in pesos. 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 expected value of $G$. Who does the game favor in the long run?

3. How much should Bruno pay Alba each time a composite number is rolled for the game to be fair ($E[G]=0$)?