Probability mass functions and probability density functions
Upcoming SlideShare
Loading in...5
×

Like this? Share it with your network

Share

Probability mass functions and probability density functions

  • 840 views
Uploaded on

 

  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Be the first to comment
    Be the first to like this
No Downloads

Views

Total Views
840
On Slideshare
840
From Embeds
0
Number of Embeds
0

Actions

Shares
Downloads
5
Comments
0
Likes
0

Embeds 0

No embeds

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
    No notes for slide

Transcript

  • 1. Probability Mass Functions and Probability Density Functions The probability mass function or pmf, fX (x) of a discrete random vari-able X is given by fX (x) =P(X = x) for all x. The probability density function or pdf, fX (x) of a continuous random xvariable X is the function that satisfies FX (x) = −∞ fX (t)∂t for all x A widely accepted convention which we will adopt, is to use an uppercaseletter for the cdf and a lowercase letter for the pmf or pdf. We must be a little more careful in our definition of a pdf in the continuouscase. If we try to naively calculate P(X = x) for a continuous random variablewe get the following: Since {X = x} ⊂ {x − < X ≤ x} for any > 0, we have from Theorem2(3), P{X = x} ≤P{x − < X ≤ x} = FX (x) − FX (x − ) for any > 0. Therefore, 0 ≤ P{X = x} ≤ lim [FX (x) − FX (x − )] = 0 by the continuity →0of FX . A note on notation: The expression “X has a distribution given by FX (x)”is abbreviated symbolically by “X ∼ FX (x),” where we read they symbol “∼”as is distributed as” or “follows”. Theorem 5: A function fX (x) is a pdf or pmf of a random variable X ifand only if: (1) fX (x) ≥ 0 for all x ∞ (2) −∞ fX (x)∂x = 1 (pdf) and X f (x) = 1 (pmf) x 1