Continuous probability distributions
How long exactly does the bus take to arrive? How long exactly does the battery in your earphones last? These questions don't have a "yes or no" answer among a few values: the time could be $14$ minutes, $14,3$ minutes, $14,37$ minutes... any number in an interval. Until now, random variables took isolated values (number of heads, number of correct answers); now we turn to the case where the variable can take any real value in an interval.
From discrete to continuous
In a discrete variable there are separate bars: between $1$ breakdown and $2$ breakdowns there is nothing in between that makes sense. In a continuous variable there is a curve with no gaps: between $14$ and $15$ minutes there are infinitely many possible values ($14,2$; $14,37$; $14,371$...).
The density function
In the discrete case, a table with $P[X=x_i]$ for each value was enough. In the continuous case that doesn't work: there are infinitely many possible values, so probability can't be shared out value by value. Instead we use a curve, the density function, and the probability of an interval is the area under that curve.
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…
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.
- a) Determine the value of the density function on that interval so that it is valid.
- b) Calculate the probability that the bus takes less than 90 minutes to complete the route.
- c) Graph the density function.
You can do it on paper and send a photo with your answer.
Shaped densities: rectangle, triangle and piecewise
The simplest density is constant on an interval (uniform density): the area is then a rectangle, $\text{base} \cdot \text{height}$. But many real situations (a reaction time, or a waiting time that rises and then falls) are modelled with a triangular density or one defined piecewise, and the area is calculated piece by piece using geometry: rectangles, triangles and trapeziums.
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…
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:
- a) The value of the maximum ordinate $k$ so that it is a valid density function.
- b) The probability that a young person takes between 15 and 20 minutes to disconnect.
- 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…
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:
$f(x) = \begin{cases} kx & \text{if } 0 \le x < 2 \\ k(4 - x) & \text{if } 2 \le x \le 4 \\ 0 & \text{otherwise} \end{cases}$
- a) Determine with reasons the value of the constant $k$ so that $f(x)$ is a probability density function.
- b) Calculate the probability that a randomly chosen student studies between 1.5 and 3 hours.
- c) What is the probability that they study for exactly 2 hours? Justify your answer based on the nature of the variable.
An exact value has no area: $P[X=a]=0$
A first look: the Gaussian curve
The most important continuous density in practice is the so-called normal distribution, whose graph is the bell curve (or Gaussian curve). It appears again and again when measuring heights, reaction times, test scores or the weights of the same product: most values cluster near the centre and there are fewer and fewer as you move away.
Two numbers are enough to describe it completely: the mean $\mu$ (where the centre is) and the standard deviation $\sigma$ (how narrow or wide the bell is). Calculating exact probabilities with it —standardising, using tables— is the content of the next section; here it is enough to recognise its shape: symmetrical, with a single maximum at $\mu$, and with total area $1$ like any density function.
Back to the beginning: the probability that the bus takes exactly $14{,}000\ldots$ minutes is $0$; with continuous quantities we cannot predict isolated instants, only the probability that the value falls within an interval, such as taking between $14$ and $15$ minutes.