2.1.2 Cumulative Distribution Functions (CDFs) Flashcards
What is a cumulative distribution function (CDF)?
A function that gives the cumulative probability of a random variable taking a value less than or equal to a given value.
How is the CDF of a random variable (X) denoted?
( F_X(x) ) or ( P(X leq x) ).
capital letter functions
How do you compute the CDF for a discrete random variable?
By summing the probabilities of all values less than or equal to ( x ):
[ F_X(x) = P(X \leq x) = \sum_{t \leq x} P(X = t) ]
What is the relationship between the PMF and the CDF?
The CDF at ( x ) is the sum of all PMF values up to ( x ), while the PMF at ( x ) is found by taking the difference:
[ P(X = x) = F_X(x) - F_X(x^-) ]
How do you compute the CDF for a continuous random variable?
By integrating the probability density function (PDF):
[ F_X(x) = \int_{-\infty}^{x} f_X(t) dt ]
How do you obtain the PDF from the CDF?
By differentiating the CDF:
[ f_X(x) = \frac{d}{dx} F_X(x) ]
What are the three main properties of a valid CDF?
- Non-decreasing: ( F_X(x) ) never decreases as ( x ) increases.
- Boundary values: ( F_X(-\infty) = 0 ) and ( F_X(\infty) = 1 ).
- Right continuity: ( F_X(x) ) is continuous from the right.
Why does ( P(X = x) = 0 ) for a continuous random variable?
Because the probability of observing any exact value in a continuous distribution is zero.
How do you find ( P(a < X \leq b) ) using the CDF?
[ P(a < X \leq b) = F_X(b) - F_X(a) ]
If given a CDF, how do you find the probability of a discrete value?
[ P(X = x) = F_X(x) - F_X(x^-) ]
where ( F_X(x^-) ) is the limit from the left.
Example: Suppose a discrete random variable ( X ) has the following CDF values: [ F_X(0) = 0.2, \quad F_X(1) = 0.6, \quad F_X(2) = 0.9, \quad F_X(3) = 1.0. ] Find ( P(X = 1) ).
[ P(X = 1) = F_X(1) - F_X(0) = 0.6 - 0.2 = 0.4. ]
