17  Continuous RVs

Definition 17.1 (Continuous random variable) We say a random variable \(X\) has a continuous random variable if the possible values of \(X\) takes the form of a continuum.

Definition 17.2 (Probability density function) For a continuous random variable \(X\), the probability density function (PDF) of \(X\) is a real-valued function \(f:\mathbb{R}\to[0,\infty)\) such that \[P(a\leq X\leq b)=\int_{a}^{b}f(x)dx.\]

Continuous random variables are usually measurements. Examples include height, weight, temperature, the amount of money and so on.

Density is not probability

PDF differs from the discrete PMF in important ways:

  • For a continuous random variable, \(P(X=x)=0\) for all \(x\);
  • The quantity \(f(x)\) is not a probability. To get the probability, we integrate the PDF (probability is the area under the PDF): \[P(a<X\leq b)=F(b)-F(a)=\int_{a}^{b}f(x)dx.\]
  • Since any single value has probability 0, including or excluding endpoints does not matter. \[P(a<X<b)=P(a<X\leq b)=P(a\leq X<b)=P(a\leq X\leq b).\]

Proposition 17.1 If \(X\) has density function \(f\) then

  1. \(P(X=x)=0\) for all \(x\in\mathbb{R}\)
  2. \(P(a\leq X\leq b) =\int_a^b f(x) dx\)
  3. \(\int_{-\infty}^{\infty}f(x)dx=1\)

Example 17.1 (Uniform distribution) A uniform distribution is a probability distribution where all values within a specified interval \([a, b]\) are equally likely to occur, and its probability density function (PDF) is given by: \[f(x) = \frac{1}{b - a} \quad \text{for} \quad a \leq x \leq b\] and \(f(x) = 0\) otherwise.

Don’t confuse a random variable with its distribution

If random variable \(X\) has distribution \(f(x)\), the distribution of \(X^2\) is not \(f^2(x)\). To get the distribution of \(X+Y\), you can’t just add up \(f_X(x)+f_Y(y)\). The right way to do it will be discussed in later chapters (transformation and convolution).

Example 17.2 Every morning, a student waits for the elevator in their dormitory. The waiting time (in minutes) is equally likely to be anywhere between 0 and 3 minutes, depending on when the elevator arrives.

  1. Define \(X\) = “the student’s elevator waiting time.” What is the support of \(X\)?
  2. Derive the probability density function (PDF) of \(X\).
  3. Compute \(P(X \leq 1)\), i.e. the probability that the waiting time is at most 1 minute.
  4. If the student must wait more than 2 minutes, they decide to take the stairs instead. Define a new indicator random variable \(Y\), which equals \(1\) if \(X>2\) and \(0\) otherwise. Compute \(P(Y=1)\).