Exercises: Continuous probability distributions

It is known that the time the school bus takes to complete its morning route follows a continuous probability distribution whose density function is constant on the interval $[80, 120]$ minutes, and equals $0$ at any other point.

1. a) Determine the value of the density function on that interval so that it is valid.

1. b) Calculate the probability that the bus takes less than 90 minutes to complete the route.

1. c) Graph the density function.

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

Observe the following two curves corresponding to normal distributions. Both represent the weight of two different breeds of domestic cats.

1. a) Do both distributions have the same mean $\mu$? Justify your answer by observing the position of the maximum.
2. b) Which of the two distributions has a larger standard deviation $\sigma$? Explain how this parameter affects the shape of the "bell curve".

Using the standard normal distribution table $Z \sim N(0, 1)$, obtain the following direct probabilities:

1. a) $P(Z < 1,25)$
2. b) $P(Z \leq 0,47)$
3. c) $P(Z < 2,03)$

Briefly explain what the value obtained in section (a) represents with respect to the total area under the Gaussian curve.

In a standard normal distribution $Z \sim N(0, 1)$, calculate the following probabilities using the standard normal table, the symmetry properties and the complementary event:

1. a) $P(Z > 1,12)$

1. b) $P(Z < -0,54)$

1. c) $P(Z \geq -1,96)$

Also, sketch the bell curve for part b), shading the area you are calculating and marking the values $-0,54$ and $0$ on the axis.

The score in a popular mental agility test for teenagers follows a normal distribution $X \sim N(100, 15)$.

1. a) Standardise the value $X = 115$ and explain what the result obtained means (how many standard deviations from the mean it is).
2. b) Standardise the value $X = 80$. What does the negative sign indicate in this case?

The battery life of a mid-range pair of wireless headphones follows a normal distribution with a mean of 6 hours and a standard deviation of 0.5 hours ($X \sim N(6; 0,5)$).

Calculate the probability that, choosing a pair of headphones at random, the battery lasts less than 7 hours. (Remember to standardise the variable first in order to use the table).

Calculate the area enclosed in the following intervals for a standard normal distribution $Z \sim N(0, 1)$:

1. a) $P(0,5 < Z < 1,5)$

1. b) $P(-1,2 < Z < 1,2)$

Graph the interval from part b) (draw the bell curve and shade the required area) and explain why that area is symmetric about the vertical axis.

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

Find the value of the x-coordinate $k$ in the standard normal distribution table $N(0, 1)$ in the following cases (inverse problem):

1. a) $P(Z < k) = 0,8413$
2. b) $P(Z < k) = 0,9750$
3. c) $P(Z < k) = 0,5000$

What happens to the value of $k$ as the probability approaches 1? Justify your answer by observing the shape of the bell curve at its tails.

Laura is analysing the duration $X$ (in hours) of calls to a citizen helpline, modelled by a uniform distribution on the interval $[0, 2]$, with probability density function $f(x) = 0,5$ if $0 \le x \le 2$ (and $0$ otherwise).

Laura writes in her notebook:

"1) Since $f(1) = 0,5$, the probability that a call lasts exactly one hour is $50\,\%$.

2) Since $P(X \le 1) = 0,5$ and $P(X = 1) = 0,5$, the probability that it lasts strictly less than one hour is $P(X < 1) = P(X \le 1) - P(X = 1) = 0,5 - 0,5 = 0$".

1. a) Identify the two conceptual errors made by Laura, explaining why she confuses the height of the density function with a probability and why she fails when calculating the point probability of a continuous variable.
2. b) Explain, using reasoned arguments based on the geometric idea of area (base times height), why for any continuous variable it holds that $P(X < 1) = P(X \le 1)$.
3. c) Correct Laura's calculations and find the correct values of $P(X = 1)$ and $P(X < 1)$.

The waiting time $X$ (in minutes) in the queue at a theatre ticket office is modelled using the triangular density function:

$f(x) = \begin{cases} \frac{1}{18}x & \text{if } 0 \le x \le 6 \\ 0 & \text{otherwise} \end{cases}$

We want to calculate the probability of waiting at least 4 minutes, that is, $P(X \ge 4)$.

Two classmates propose different approaches:

• Sergio suggests calculating the area under the line between $x = 4$ and $x = 6$ directly, identifying the region as a trapezoid with bases $f(4)$ and $f(6)$ and height $2$.

• Elena proposes calculating the probability of the complementary event $P(X < 4)$ as the area of a triangle with base $4$ and height $f(4)$, and then subtracting: $1 - P(X < 4)$.

1. a) Compare both procedures by analysing the geometric formulas each requires. Which one do you consider more direct or less prone to arithmetic error? Justify your answer.
2. b) Choose one of the two methods and calculate the exact numerical value of $P(X \ge 4)$ as an irreducible fraction.

The time taken by a group of young people to completely disconnect after turning off their mobile phones before sleeping is being studied. The density function of this continuous random variable, $x$, expressed in minutes, is given by the following graph:

Calculate with reasons:

1. a) The value of the maximum ordinate $k$ so that it is a valid density function.
2. b) The probability that a young person takes between 15 and 20 minutes to disconnect.
3. c) What is the probability that they take exactly 22 minutes?

Using the standard normal distribution table $N(0, 1)$, calculate the following probabilities:

1. a) $P(Z < 1,48)$
2. b) $P(Z > 0,72)$
3. c) $P(Z < -1,15)$
4. d) $P(-0,5 < Z < 1,2)$

Furthermore, find the value of the positive constant $a$ such that $P(-a < Z < a) = 0,90$.

Two friends, Lucía and Marco, play for different esports teams. Lucía's score in the game "Galaxy Quest" follows a $N(1200, 15)$ distribution, while Marco's in "Pixel Arena" follows a $N(850, 10)$ distribution.

In the last game, Lucía scored 1 235 points and Marco scored 875 points. Use standardisation of variables to determine which of the two achieved a more meritorious result compared to their group.

The height of Year 12 students at a secondary school follows a normal distribution with mean $172$ cm and standard deviation $8$ cm. If we select a student at random, find the probability that:

1. a) Their height is greater than $185$ cm.
2. b) Their height is less than $160$ cm.
3. c) Their height is between $165$ and $180$ cm.

A company that manufactures school backpacks has verified that the weight they support before the seams break follows a normal distribution with a mean of $24$ kg and a standard deviation of $3,5$ kg.

1. a) What percentage of backpacks support a weight between $20$ and $28$ kg?
2. b) If the company manufactures a batch of $5000$ backpacks, how many are expected to fail before $18$ kg?

On a music streaming platform, the time users spend listening to a "Top 50" playlist follows a $N(45, 12)$ minutes distribution. The platform decides to send a satisfaction survey to the top $15\%$ of users who spend the most time listening to this playlist. After how many minutes of listening will a user receive the survey?

The weight of chocolate bars produced by a factory follows a normal distribution with a mean of $200$ grams. Quality control rejects those bars that weigh less than $195$ grams. Knowing that $3\%$ of the bars are rejected, calculate the value of the standard deviation for this production.

AEMET records the daily precipitation $X$ (in litres per square metre) at a weather station on rainy days using the following density function on the interval $[0, 10]$:

$f(x) = \begin{cases} \frac{1}{50}(10 - x) & \text{if } 0 \le x \le 10 \\ 0 & \text{otherwise} \end{cases}$

1. a) Before performing numerical area calculations, make a mental sketch of the graph (a decreasing straight line from $(0; 0,2)$ to $(10, 0)$) and estimate qualitatively with justification:

• Which probability do you expect to be greater: that it rains between 0 and 2 litres ($P(0 \le X \le 2)$) or between 8 and 10 litres ($P(8 \le X \le 10)$)?

• Knowing that the total interval has width 10, do you think the probability that it rains at most 5 litres, $P(X \le 5)$, will be greater than, equal to, or less than $0,5$? Explain your deduction by looking at how the area under the graph is distributed.

1. b) Verify your estimates by calculating the values of $P(0 \le X \le 2)$, $P(8 \le X \le 10)$, and $P(X \le 5)$ exactly using geometric areas (triangles or trapezoids).
2. c) Compare the numerical results obtained with your predictions from part a) and conclude whether your intuition was correct.

According to urban mobility data on electric vehicle charging, the time $X$ (in hours) that a car stays at a public semi-fast charging station is modelled as a continuous random variable on the interval $[0, 4]$. The proposed density function has two linear phases:

$f(x) = \begin{cases} c \cdot x & \text{if } 0 \le x < 1 \\ k \cdot (4 - x) & \text{if } 1 \le x \le 4 \\ 0 & \text{otherwise} \end{cases}$

1. a) For the function to be continuous at the transition point $x = 1$, the two branches must take the same value, that is, $c \cdot 1 = k \cdot (4 - 1)$. Use this relationship and the property that the total area under the curve must equal $1$ to determine the exact values of the constants $c$ and $k$ with justification.
2. b) The city council wants to offer a discount to users who vacate the charging bay before 2 hours. Model the situation by calculating the probability that a user receives this discount, that is, find $P(X \le 2)$ by adding the corresponding geometric areas.
3. c) If a driver claims that the probability of charging for exactly 1 hour is zero ($P(X = 1) = 0$), does that mean it is impossible for a car to disconnect precisely at the one-hour mark? Explain the difference between the mathematical definition of probability in continuous variables and the physical possibility of the event.

Consider the piecewise continuous probability function of a random variable $X$, which represents the time (in hours) that a group of teenagers spends studying mathematics over a weekend:

$f(x) = \begin{cases} kx & \text{if } 0 \le x < 2 \\ k(4 - x) & \text{if } 2 \le x \le 4 \\ 0 & \text{otherwise} \end{cases}$

1. a) Determine with reasons the value of the constant $k$ so that $f(x)$ is a probability density function.
2. b) Calculate the probability that a randomly chosen student studies between 1.5 and 3 hours.
3. c) What is the probability that they study for exactly 2 hours? Justify your answer based on the nature of the variable.

The latency (ping) in a popular online video game follows a normal distribution $N(\mu, \sigma)$. After taking thousands of measurements, the developers have observed that $15,87\%$ of the games have a latency greater than $110$ ms, while $6,68\%$ of the games have a latency of less than $75$ ms.

1. a) Calculate the parameters $\mu$ (mean) and $\sigma$ (standard deviation) of this distribution.
2. b) If a player considers a game "unplayable" when the latency exceeds $125$ ms, what percentage of games will fall into this category?

A tech company manufactures precision sensors for drones. The margin of error in measuring height with these sensors follows a normal distribution with a mean of $0$ cm and a standard deviation of $1,5$ cm. For safety reasons, the company rejects all sensors whose absolute error is greater than a critical value $k$.

1. a) If it is known that $10\%$ of the total production is rejected, calculate the value of said margin of error $k$.
2. b) What is the probability that, when choosing an accepted (non-rejected) sensor, its error is positive?

A music streaming app has launched a new recommendation algorithm. It is estimated that the probability of a user liking a recommended song is $65\%$. If in one day the algorithm recommends songs to a group of $400$ users:

1. a) Justify whether it is possible to approximate the distribution of the number of users who like the recommendation using a normal distribution.
2. b) Using the appropriate approximation (with continuity correction), calculate the probability that the recommendation is successful for a number of users between $250$ and $275$ (inclusive).

At a prestigious design school, the applicants' final marks follow a normal distribution. It is known that the mean mark is $6,8$ and that $33\%$ of the students obtain a mark higher than $7,5$.

1. a) Find the standard deviation of the marks.
2. b) If obtaining an excellence scholarship requires being in the top $5\%$ of marks, what is the minimum mark required to get said scholarship?

The weight of recycled plastic bottles collected at a sustainability centre follows a $N(120, 15)$ grams distribution. The centre classifies the bottles into three categories: "Light" (the bottom $20\%$ by weight), "Standard" (the central $60\%$) and "Heavy" (the top $20\%$ by weight).

1. a) Determine the weights that delimit each of the three categories.
2. b) If we choose $5$ bottles at random, what is the probability that exactly $3$ of them are classified as "Standard"?

Two electric scooter models are being compared. Model A has an average range of $40$ km with a standard deviation of $4$ km. Model B has an average range of $42$ km with a standard deviation of $8$ km. Both ranges follow normal distributions.

1. a) If a user needs to make a journey of $48$ km without recharging, which model offers greater reliability (higher probability of success)? Justify your calculations.
2. b) For what distance is the probability of both models failing the same?

A logistics company analyses the delivery time of its local packages. The delivery time follows a normal distribution with a standard deviation of $12$ minutes. It is known that $75\%$ of the packages are delivered in under $50$ minutes.

1. a) Calculate the mean delivery time.
2. b) The company guarantees a postage refund if the package takes more than an hour ($60$ minutes) to arrive. If $5000$ packages are sent in a month, how many refunds is the company estimated to have to make?
3. c) If the company wants to reduce the number of refunds by half by improving logistics while keeping the same mean, what should the new standard deviation be?

A logistics company based in Zaragoza analyses the time $X$ (in days) it takes to deliver an urgent domestic order. Analysts know that $X$ is continuously and uniformly distributed over an unknown interval $[a, b]$, with $0 < a < b$, so that its density function is constant: $f(x) = \frac{1}{b - a}$ for $x \in [a, b]$.

Neither the endpoint $a$ nor $b$ is known, but two verified pieces of data are available:

1. The average delivery time (the midpoint of the interval) is $3$ days.

2. The probability that a parcel takes more than 4 days to reach the recipient is exactly $25\,\%$ ($0,25$).

1. a) Set up a system of equations from the geometric conditions of the problem and deduce the exact values of $a$ and $b$ with justification.
2. b) Write the final analytical expression of the density function $f(x)$ and calculate the probability that a delivery takes less than 2 days.
3. c) Company management wants to set an advertising commitment: guarantee a maximum delivery time $T$ such that only $5\,\%$ of parcels fail to meet the guarantee (that is, $P(X > T) = 0,05$). Calculate the exact value of $T$.
4. d) If, instead of a uniform model, a normal distribution $N(3, \sigma)$ with the same mean were used, discuss whether an absolute maximum time guaranteed at $100\,\%$ would theoretically exist and explain how the shape of the density function would change at the tails.

In the university entrance exams (EvAU/EBAU) of a certain autonomous community, the grades obtained by students are modelled using a normal distribution $N(\mu, \sigma)$. The technical committee compares the results of two different sittings in which the mean grade was identical, but the dispersion was different:

• Sitting A: $X_A \sim N(6,2;\, 1,4)$

• Sitting B: $X_B \sim N(6,2;\, 0,8)$

1. a) Mentally sketch and compare the density functions of both distributions:

• At what value of $x$ does each bell curve reach its maximum, and which curve has a higher peak? Justify your answer taking into account that the area under both functions must equal $1$.

• Which of the two curves is wider, and what mathematical parameter determines that width?

1. b) To access the Medicine degree, a very demanding cut-off mark is required: at least $8,6$ points. Notice that the difference between $8,6$ and the common mean $6,2$ is $2,4$ points. Without needing to consult numerical tables, calculate how many standard deviations ($\sigma$) from the mean this cut-off mark is in Sitting A and Sitting B, and reason in which of the two sittings there was a higher percentage of students who scored above that mark.
2. c) A school counsellor claims: "In Sitting B it was easier to obtain an intermediate grade, in the interval $[5;\, 7,4]$, than in Sitting A". Critically analyse this statement: express that interval in terms of distances to the mean for each distribution and justify mathematically whether the counsellor is right.