43  Definition revisited

Continuous random variables, in many ways, are more versatile and useful than discrete distributions. One key reason is that many quantities in the physical world, such as temperature, height, weight, and time, are inherently continuous in nature. Additionally, the probability density functions (PDFs) of continuous distributions are often defined by smooth, differentiable functions. This mathematical structure allows us to apply calculus for analysis.

Definition 43.1 A random variable has a continuous distribution if its CDF is differentiable.

Remark.

• For a continuous random variable, \(P(X=x)=0\) for all \(x\);

• The density function \(f(x)\) 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. \[P(a<X<b)=P(a<X\leq b)=P(a\leq X<b)=P(a\leq X\leq b).\]

• The PDF of a continuous random variable satisfies the property: \[\int_{-\infty}^{\infty}f(x)dx=1.\]

• The CDF is the integral of PDF:

  \[F(X)=\int_{-\infty}^{x}f(x)dx,\quad f(x)=F'(x).\]

Definition 43.2 The expectation of a continuous random variable \(X\) with PDF \(f\) is \[E(X)=\int_{-\infty}^{\infty}xf(x)dx.\]

Theorem 43.1 If \(X\) is a continuous random variable with PDF \(f\) and \(g:\mathbb{R}\to\mathbb{R}\). The LOTUS applies \[E[g(X)]=\int_{-\infty}^{\infty}g(x)f(x)dx.\]

	Discrete	Continuous
PMF/PDF	\(P(X=x)=p(x)\)	\(P(a\leq X\leq b)=\int_{a}^{b}f(x)dx\)
CDF	\(F(x)=P(X\leq x)=\sum_{k\leq x}p(k)\)	\(F(x)=P(X\leq x)=\int_{-\infty}^{x}f(t)dt\)
Expectation	\(E(x)=\sum_{x}xP(X=x)\)	\(E(X)=\int_{-\infty}^{+\infty}xf(x)dx\)
LOTUS	\(E[g(x)]=\sum_{x}g(x)P(X=x)\)	\(E[g(x)]=\int_{-\infty}^{+\infty}g(x)f(x)dx\)