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.

The time taken by a group of young people to completely disconnect after turning off their cell 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?

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.

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.

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.

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 $P(-a < Z < a) = 0.90$.

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?

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?

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.

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?

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).

The height of 12th-grade students at a high 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?

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.

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 recycled plastic bottles collected at a sustainability center follows a $N(120, 15)$ grams distribution. The center 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"?

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.

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?

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?

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 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).