Exercises: Continuous density function and cumulative distribution function

The density function of a continuous random variable $X$ is defined, representing the waiting time (in minutes) for a game to start on a video game server:

$f(x) = \begin{cases} k & \text{if } 0 \le x \le 5 \\ 0 & \text{en el resto} \end{cases}$

1. a) Determine the value of the constant $k$ so that $f(x)$ is a valid density function.

1. b) Graph the function $f(x)$ using the value of $k$ you have obtained.

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

The battery life of a new wireless headphone model follows a uniform distribution between 4 and 10 hours. Its density function is constant in that interval: $f(x) = \frac{1}{6}$. Calculate the probability that the battery lasts between 5 and 8 hours, interpreting it as the area of a rectangle.

Given a standard normal distribution $Z \sim N(0, 1)$, use the probability table to find the following values directly:

• a) $P(Z \le 0,45)$

• b) $P(Z \le 1,2)$

• c) $P(Z \le 2,38)$

For a variable $Z \sim N(0, 1)$, apply the properties of the normal curve (symmetry and total area) to calculate the following probabilities:

• a) $P(Z > 1,52)$

• b) $P(Z < -0,84)$

• c) $P(Z \ge -1,15)$

Calculate the probability that the standard normal variable $Z$ takes a value lying in the interval $(0,62 \le Z \le 1,96)$. Remember that to calculate the area of an interval, you must subtract the cumulative areas: $P(a \le Z \le b) = P(Z \le b) - P(Z \le a)$.

The scores on an English placement test for Sixth Form students follow a normal distribution with a mean of $\mu = 75$ and a standard deviation of $\sigma = 8$. Standardise the following values of variable $X$ to find their corresponding value in $Z$:

• a) A score of $83$.

• b) A score of $67$.

• c) A score of $75$.

The weight of bags of crisps produced by a machine follows a distribution $N(150, 5)$ grams. Calculate the probability that a randomly chosen bag weighs less than $160$ grams. To do this, standardise the variable first and look up the result in the table.

Use the $N(0, 1)$ distribution table in reverse to find the value of the constant $k$ in the following cases:

• a) $P(Z \le k) = 0,8413$

• b) $P(Z \le k) = 0,9750$

• c) $P(Z \le k) = 0,5000$

Given the following density function of a continuous random variable $X$, representing the waiting time in minutes in a fairground ride queue before getting on:

Calculate the value of the constant $k$ so that the area under the function is equal to $1$ and graphically find the probability that a visitor waits less than $2$ minutes.

A continuous random variable $X$ models the waiting time (in minutes) at an urban bus stop and has the following probability density function:

$f(x) = \begin{cases} 0,1 & \text{if } 0 \le x \le 10 \\ 0 & \text{otherwise} \end{cases}$

Marcos needs to solve two questions about this service and writes on his exam:

"1) Since the function is $0,1$ over the entire interval, the probability of waiting between 3 and 7 minutes is $P(3 \le X \le 7) = f(7) - f(3) = 0,1 - 0,1 = 0$.

2) Moreover, since $f(5) = 0,1$, the probability of waiting exactly 5 minutes is $10\,\%$."

Identify with reasoning the two conceptual errors Marcos has made, explaining which theoretical property of continuous random variables he violates in each case. Then, calculate the correct value of $P(3 \le X \le 7)$ using the cumulative distribution function $F(x)$.

The duration of a brief technical query on a web service (in minutes) follows a continuous random variable $X$ with probability density function:

$f(x) = \begin{cases} \frac{x}{8} & \text{if } 0 \le x \le 4 \\ 0 & \text{otherwise} \end{cases}$

1. a) Before carrying out any algebraic calculations, analyse the geometric shape of the graph of $f(x)$ (a triangle increasing from 0 to 4). Make a reasoned qualitative estimate: will the median $M$ of this distribution be less than 2, exactly equal to 2, or greater than 2? Justify your conjecture by indicating where the larger proportion of the area is concentrated.
2. b) Find the analytical expression of the cumulative distribution function $F(x)$ for $x \in [0, 4]$ by calculating the area of the accumulated triangle from 0 to a generic abscissa $x$.
3. c) Solve the equation $F(M) = 0,5$ to find the exact value of the median (round to two decimal places) and check whether your initial estimate from part a) was correct.

The reaction time, in seconds, of a group of teenagers to a visual stimulus in a video game is being studied. The density function of this continuous variable is given by the following expression:

$f(x) = \begin{cases} kx & \text{if } 0 \le x \le 2 \\ 0 & \text{otherwise} \end{cases}$

Determine:

1. a) The value of the constant $k$ so that $f(x)$ is a valid density function.
2. b) The probability that a randomly chosen player has a reaction time greater than $1,5$ seconds.
3. c) The probability that the reaction time is between exactly $0,5$ and $1$ seconds.

The waiting time, in minutes, for an update to download on a console follows a continuous distribution whose density function is in the shape of a trapezium. The function is defined between $0$ and $10$ minutes. Between minutes $0$ and $4$, the function is constant with a value of $h$, and between minutes $4$ and $10$ it decreases linearly until reaching $0$.

1. a) Calculate the value of $h$ so that it is a density function.
2. b) Calculate the probability that the update takes less than $6$ minutes to download.
3. c) If a user has been waiting for more than $4$ minutes, what is the probability that the download finishes before minute $8$?

Given a random variable $Z$ following a standard normal distribution $N(0, 1)$, use symmetry properties and the total area to calculate the following probabilities:

1. a) $P(Z < 1,42)$
2. b) $P(Z > -0,75)$
3. c) $P(1,12 < Z < 2,05)$
4. d) $P(-1,5 < Z < 1,5)$
5. e) Find the value of $k$ such that $P(Z < k) = 0,8830$

The heights of 11th-grade students in a city follow a normal distribution with a mean of $170$ cm and a standard deviation of $8$ cm.

1. a) What percentage of students are between $162$ and $178$ cm tall?
2. b) If a sports team selection requires a minimum height of $154$ cm, what is the probability that a randomly chosen student does not meet this requirement?
3. c) What is the probability that a student is taller than $180$ cm?

The scores obtained by students at a school in the university entrance exam follow a normal distribution $N(6,8 ; 1,2)$.

1. a) If entering a highly oversubscribed degree course requires a score higher than $8,5$, what percentage of the school's students would have the chance to be admitted?
2. b) The school decides to award an honours mention to the top $5 \%$ of students with the highest marks. From what score onwards is this mention awarded?
3. c) What is the probability that a student has a mark lower than $5$?

A courier company has analysed the weight of the packages it sends daily. It has been shown that the weight follows a normal distribution. We know that $15,87 \%$ of the packages weigh less than $2$ kg and that $2,28 \%$ of the packages weigh more than $8$ kg.

1. a) Determine the mean and standard deviation of the weight of the packages.
2. b) What is the probability that a randomly chosen package weighs between $3$ and $6$ kg?

In a study on monthly leisure spending by young people in a city, it has been determined that it follows a normal distribution with a mean of $80$ € and a standard deviation of $15$ €.

1. a) Calculate the interval centred on the mean that includes $95 \%$ of young people.
2. b) If we select a young person at random, what is the probability that their monthly expenditure does not differ from the mean by more than $10$ €?
3. c) If there are $5.000$ young people in the city, how many of them are estimated to spend more than $110$ € a month?

A farmer classifies the oranges from his harvest by their diameter. The diameter follows a normal distribution $N(8 ; 0,6)$ in centimetres.

1. a) Oranges with a diameter of less than $7$ cm are sent for juice manufacturing. What is the probability that an orange ends up at the juice factory?
2. b) "Extra" oranges are those with a diameter between $8,5$ and $9,5$ cm. What percentage of the harvest is labelled as "Extra"?
3. c) If the farmer wants to set aside $10 \%$ of the largest oranges for luxury export, what must be the minimum diameter of these oranges?

According to punctuality records on a Renfe commuter rail line, the delay $X$ (in minutes) of a train follows a decreasing probability density function:

$f(x) = \begin{cases} k \cdot (6 - x) & \text{if } 0 \le x \le 6 \\ 0 & \text{otherwise} \end{cases}$

1. a) Determine the exact value of the constant $k$ using the area of the triangular region enclosed by $f(x)$ and the x-axis to ensure that it is a valid probability density function.
2. b) Deduce the cumulative distribution function $F(x)$ for the interval $[0, 6]$ by accumulating the area under the curve from the origin up to a delay $x$ (you can calculate the area of the trapezoid formed or subtract from the total area of 1 the triangle remaining to the right of $x$).
3. c) The company applies a voluntary compensation policy that refunds the ticket price if the delay exceeds 4 minutes. Using $F(x)$, calculate the probability that a randomly chosen passenger is entitled to compensation and express the result as a percentage.

Let $f(x)$ be a piecewise function defined as follows:

$f(x) = \begin{cases} k \cdot x & \text{if } 0 \le x < 2 \\ k \cdot (4 - x) & \text{if } 2 \le x \le 4 \\ 0 & \text{en el resto} \end{cases}$

1. a) Determine, with justification, the value of the constant $k$ so that $f(x)$ is a density function of a continuous random variable $X$.

1. b) Once $k$ has been found, calculate the probability that the variable takes a value in the interval $[1, 3]$, that is, $P(1 \le X \le 3)$.

1. c) Graph the function $f(x)$ and shade in the drawing the area corresponding to the probability calculated in part b).

The daily usage time of a map app on the mobile phones of delivery drivers in a city follows a normal distribution $N(\mu, \sigma)$. It is known that $15,87 \%$ of drivers use it for less than $40$ minutes a day, while $2,28 \%$ use it for more than $80$ minutes. Calculate the parameters $\mu$ and $\sigma$ of this distribution. What is the probability that a randomly chosen driver uses the app for between $50$ and $70$ minutes?

An electronic component manufacturing company claims that $4 \%$ of its touchscreens have some factory micro-defect. If a school purchases a batch of $250$ screens for its classrooms, calculate the probability that:

1. a) There are exactly $12$ defective screens.
2. b) There are between $8$ and $15$ defective screens (inclusive).

Note: Justify whether it is possible to approximate the binomial distribution to the normal distribution and apply Yates' continuity correction.

In a young musical talent competition, the scores follow a normal distribution with a mean of $72$ points. It is known that to obtain an "Honourable Mention", a score is required such that only $10 \%$ of participants achieve it. If the minimum score to receive this mention is $84,8$ points, what is the standard deviation of the scores? If $500$ participants enter, how many are expected to have a score below $60$ points?

The lifespan of a model of LED light bulbs follows a normal distribution with a mean of $30$ thousand hours. If it is known that the probability of a bulb lasting more than $34$ thousand hours is the same as the probability of it lasting less than $26$ thousand hours, and both are equal to $0,1587$, calculate the variance of the distribution. If a local council needs the streetlights in a park to function for $33$ thousand hours without replacement, what is the probability that a bulb will blow before reaching that time?

It has been established that the weight of $500$-gram packets of coffee coming from a packaging machine follows a normal distribution $N(505, 10)$. Quality control rejects packets weighing less than $490$ grams or more than $525$ grams.

1. a) What percentage of packets is rejected by the factory?
2. b) If the manager decides to adjust the machine so that the mean weight is $\mu$, whilst keeping the standard deviation $\sigma = 10$, what value should $\mu$ have so that the risk of a packet weighing less than $500$ grams is only $2,5 \%$?

Given the density function $f(x) = \frac{1}{8}(x-1)$ defined for the interval $x \in [1, 5]$ and $f(x)=0$ elsewhere:

1. a) Check that the total area under the curve is indeed $1$.
2. b) Find a value $M$ (median) such that $P(X \le M) = 0,5$.
3. c) Calculate the conditional probability $P(X > 4 | X > 2)$.

On a streaming platform, the time users watch a series before abandoning it follows a normal distribution. It is known that the mean time is $45$ minutes. If $25 \%$ of users watch for more than $60$ minutes, calculate the standard deviation. Determine the symmetrical interval about the mean $(45 - d, 45 + d)$ containing $95 \%$ of the viewing times of users.

Consider a continuous random variable $X$ whose symmetric probability density function is given by:

$f(x) = \begin{cases} x & \text{if } 0 \le x < 1 \\ 2 - x & \text{if } 1 \le x \le 2 \\ 0 & \text{otherwise} \end{cases}$

Two different calculation scenarios are posed:

1. a) Scenario 1 (single manual calculation): You must obtain $P(X \ge 0,3)$. Compare two methods: calculating the direct area by dividing the region into a trapezoid on $[0,3; 1]$ plus the triangle on $[1, 2]$, versus calculating it using the complementary event $1 - P(X < 0,3)$ using the area of the left tail. Justify which of the two approaches is more efficient and use it to give the exact result.
2. b) Scenario 2 (algorithmic design): A mobile app needs to respond instantly to more than 10\\,000 daily requests of the form $P(X \le x)$ for values of $x$ scattered throughout the interval $[0, 2]$. Justify why in this case it is preferable to deduce the cumulative distribution function $F(x)$ piecewise beforehand instead of calculating the geometric areas on $f(x)$ for each query.
3. c) Analytically determine the complete expression of $F(x)$ for all $\mathbb{R}$ and prove its continuity at the transition point $x = 1$.

In the admission and triage service of a local hospital, the patient classification time $X$ (measured in tens of minutes, with a range from 0 to 20 minutes, that is, $x \in [0, 2]$) is modelled through its cumulative distribution function:

$F(x) = \begin{cases} 0 & \text{if } x < 0 \\ k \cdot x^2 & \text{if } 0 \le x \le 1 \\ a \cdot x - 0,25 & \text{if } 1 < x \le 2 \\ 1 & \text{if } x > 2 \end{cases}$

1. a) Reason which mandatory theoretical properties of any continuous cumulative distribution function (boundary conditions at the endpoints of the range and absence of jumps) allow setting up a system to uniquely determine the values of the parameters $k$ and $a$. Deduce these values.
2. b) The physician on duty claims: "Since the maximum time set in the protocol is 20 minutes ($x = 2$), the median triage time must necessarily be half of that: 10 minutes ($x = 1$)". Mathematically prove whether this claim is true or false by calculating the exact value of the median $M$ in minutes.
3. c) Analyse how the rate of increase of $F(x)$ varies in the first interval $[0, 1]$ compared to the second interval $(1, 2]$, and interpret in the hospital context whether most consultations are concentrated in short or long times.