A continuous random variable is a random variable XX for which P(X=x)=0P(X=x)=0 for every point x∈R, but there are intervals (a,b) over which P(X∈(a,b))>0.
Example: Random events that happen in real time are represented using continuous random variables. For example, let T be the amount of time after 12:00pm until a company gets it's first customer service phone call for the day. The probability that the first phone call takes exactly 1 minute is 0, because it is impossible to measure so precisely. What if the first phone call comes at 1 minute and 0.0000000001 seconds? However, it is possible to measure whether the first phone call comes in between 1 minute and 1 minute 5 seconds.
The probability density fucntion (pdf) of a continuous random variable X is similar to the pmf, but sums are replaced by integrals.
The pdf of X is a function f such that
P(X∈A)=∫Af(x)dx
Example: Let X be a continuous random variable with pdf f(x)=12x for 0≤x≤2. Find the probability that 1<X<2.
The event {1<X<2} is equivalent to the event {X∈(1,2)}, so we can directly compute:
P(X∈(1,2))=∫(1,2)f(x)dx=∫2112xdx=14x2|21=14⋅22−14⋅12=34
The pdf represents a way to compute probabilities, so there are certain properties that it must satisfy.
Check your understanding:
Let X be a continuous random variable with pdf f(x) defined by f(x)=2+x3 for −1<x<0 and f(x)=2−x3 for 0≤x<1.
1. What is P(X=0)?
Unanswered
2. What is P(X<0)?
Unanswered
3. What is P(X>0.5)?
Unanswered
4. What is P(|X|<0.25)?
Unanswered