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Definition

A continuous random variable is a random variable XX for which P(X=x)=0P(X=x)=0 for every point xR, 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.

Probability Density Function (pdf)

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(XA)=Af(x)dx




Example: Let X be a continuous random variable with pdf f(x)=12x for 0x2. 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=14221412=34

Properties of the pdf

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)=2x3 for 0x<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