Continuous Density Functions and the Normal Distribution

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

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

Determine the value of the constant $k$ for $f(x)$ to be a valid probability density function and graph the function.

The battery life of a new model of wireless headphones follows a uniform distribution between 4 and 10 hours. Its probability 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.

The reaction time, in seconds, of a group of teenagers to a visual stimulus in a video game is being studied. The probability 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 $k$ for $f(x)$ to be a valid probability 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 exactly between $0,5$ and $1$ seconds.

The waiting time, in minutes, until an update is downloaded on a console follows a continuous distribution whose probability density function has the shape of a trapezoid. 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 $h$ for it to be a probability 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 $8$?

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

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

Determine, justifying your answer, the value of the constant $k$ for $f(x)$ to be a probability density function of a continuous random variable $X$. Once $k$ is found, calculate the probability that the variable takes a value in the interval $[1, 3]$ and represent it graphically.

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

1. a) Show 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 $P(X > 4 | X > 2)$.

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) $P(Z \le 2,38)$

Use the table of the $N(0, 1)$ distribution 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) $P(Z \le k) = 0,5000$

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 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: $P(a \le Z \le b) = P(Z \le b) - P(Z \le a)$.

Given a random variable $Z$ that follows a standard normal distribution $N(0, 1)$, use the properties of symmetry 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 scores of an English proficiency exam for high school seniors follow a normal distribution with mean $\mu = 75$ and standard deviation $\sigma = 8$. Standardize the following values of the 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 the bags of potato chips produced by a machine follows a distribution of $N(150, 5)$ ounces. Calculate the probability that a randomly chosen bag weighs less than $160$ ounces. To do this, first standardize the variable and look up the result in the table.

The battery life of a specific model of wireless headphones follows a normal distribution with a mean of $30$ hours and a standard deviation of $4$ hours.

1. a) What percentage of headphones have a battery that lasts between $26$ and $34$ hours?
2. b) If the manufacturer guarantees that the headphones last at least $22$ hours, what is the probability of finding headphones that do not meet the guarantee?
3. c) What is the probability that the battery lasts more than $35$ hours?

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

1. a) If accessing a highly demanded major requires a score higher than $8,5$, what percentage of the school's students would have the option to enter?
2. b) The school decides to award an honorable mention to the $5 \%$ of students with the best scores. From what score is this mention received?
3. c) What is the probability that a student has a score lower than $5$?

In a study on the monthly leisure expenditure of 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 centered on the mean that comprises $95 \%$ of the young people.
2. b) If we select a young person at random, what is the probability that their monthly expenditure does not deviate 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 inches.

1. a) Oranges with a diameter of less than $7$ inches are destined for juice manufacturing. What probability does an orange have of ending up in the juice factory?
2. b) "Extra" oranges are those whose diameter is between $8,5$ and $9,5$ inches. What percentage of the harvest is labeled as "Extra"?
3. c) If the farmer wants to separate the largest $10 \%$ of the oranges for a luxury export, what must be the minimum diameter of these oranges?

In a young musical talent contest, the scores follow a normal distribution with a mean of $72$ points. It is known that to obtain an "Honorable Mention," a score is required such that only $10 \%$ of the 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 lower than $60$ points?

It has been verified that the weight of the $500$-ounce coffee bags leaving a packaging machine follows a normal distribution $N(505, 10)$. Quality control rejects those bags that weigh less than $490$ ounces or more than $525$ ounces.

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

On a streaming platform, the time users watch a show 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 interval symmetrical with respect to the mean $(45 - d, 45 + d)$ that contains $95 \%$ of the users' viewing times.

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

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

The daily usage time of a map application on the cell phones of delivery drivers in a city follows a normal distribution $N(\mu, \sigma)$. It is known that $15,87 \%$ of the delivery 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 delivery driver uses the application between $50$ and $70$ minutes?

The battery life of a model of wireless headphones follows a normal distribution with a mean of $30$ hours. If it is known that the probability of a battery lasting more than $34$ hours is the same as the probability of it lasting less than $26$ hours, and both are $0,1587$, calculate the variance of the distribution. If a user needs the headphones to work during a $33$-hour trip without recharging, what is the probability that the battery runs out before reaching their destination?

An electronic component manufacturing company claims that $4 \%$ of its touch screens have some factory micro-defect. If a school acquires 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's continuity correction.