Discrete probability function and 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, $P(X \ge 2)$.

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 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.

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.

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.

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 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.