Continuous Density and Distribution Functions

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.

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.

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.

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 $8$?

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

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

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).

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 $P(X > 4 | X > 2)$.

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) $P(Z \ge -1.15)$

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 $P(Z < k) = 0.8830$

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

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) $P(Z \le k) = 0.5000$

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 chips 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, standardize the variable first and look up the result in the table.

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?

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?

The daily usage time of a map app on the cell 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?

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?

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.