Continuous density function and distribution function
You already know that with a continuous variable the probability of an interval is an area under $f(x)$. But calculating a different area each time is slow. The distribution function solves this once and for all: for each $x$, it accumulates all the probability from the left-hand end up to that point. With it, any probability can be found by subtracting two values, without redoing the area each time.
The distribution function for a continuous variable
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:…
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:
$f(x) = \begin{cases} k & \text{if } 0 \le x \le 5 \\ 0 & \text{en el resto} \end{cases}$
- a) Determine the value of the constant $k$ so that $f(x)$ is a valid density function.
- 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.
From density to distribution: accumulated area
$f(x)$ and $F(x)$ are two ways of looking at the same thing: $f$ is the curve whose area gives the probability; $F$ is already that area, accumulated up to each point. That is why, to calculate the probability of any interval, it is enough to subtract two values of $F$:
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:…
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…
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:
- a) The value of the constant $k$ so that $f(x)$ is a valid density function.
- b) The probability that a randomly chosen player has a reaction time greater than $1,5$ seconds.
- c) The probability that the reaction time is between exactly $0,5$ and $1$ seconds.
Building F(x) step by step
When $f(x)$ is defined piecewise, $F(x)$ is also built piecewise: in each section, $F(x)$ is the area accumulated from the start up to that $x$, using the appropriate geometric shape (rectangle, triangle, trapezium).
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…
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 $0$.
- a) Calculate the value of $h$ so that it is a density function.
- b) Calculate the probability that the update takes less than $6$ minutes to download.
- 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:…
Let $f(x)$ be a piecewise function defined as follows:
$f(x) = \begin{cases} k \cdot x & \text{if } 0 \le x < 2 \\ k \cdot (4 - x) & \text{if } 2 \le x \le 4 \\ 0 & \text{en el resto} \end{cases}$
- a) Determine, with justification, the value of the constant $k$ so that $f(x)$ is a density function of a continuous random variable $X$.
- b) Once $k$ has been found, calculate the probability that the variable takes a value in the interval $[1, 3]$, that is, $P(1 \le X \le 3)$.
- c) Graph the function $f(x)$ and shade in the drawing the area corresponding to the probability calculated in part b).
The median: the point that leaves half on each side
Given the density function $f(x) = \frac{1}{8}(x-1)$ defined for the interval $x \in [1, 5]$ and $f(x)=0$ elsewhere: a) Check that the total area under the curve is…
Given the density function $f(x) = \frac{1}{8}(x-1)$ defined for the interval $x \in [1, 5]$ and $f(x)=0$ elsewhere:
- a) Check that the total area under the curve is indeed $1$.
- b) Find a value $M$ (median) such that $P(X \le M) = 0,5$.
- c) Calculate the conditional probability $P(X > 4 | X > 2)$.