Some Useful Random Variables
1) Flip a coin # of H
2) Manufacture a Chip # of acceptable chips
3) Bits you transmit successfully by a modem
Geometric Random Variable
Number of trials until (and including) a success for an underlying Bernoulli
1) Repeated coin flips # of tosses until H
2) Manufacture chips 3 of chips produced until an acceptable time
"# of successes in n trials"
1) Flip a coin n times. # of heads.
2) Manufacture n chips. # of acceptable chips.
Note: Binomial where are independent Bernoulli trials
Note: n=1; Binomial=Bernoulli;
"number of trials until (and including) the kth success with an underlying Bernoulli"
where is successes in trials
Note: Pascal where are geometric R.V.
Note: K=1 Pascal=Geometric
# of flips until the kth H
Discrete Uniform R.V.
1) Rolling a die.
2) Flip a fair coin. =# of H
(Exercise) limiting case of binomial with
PMF is a complete model for a random variable
Cumulative Distribution Function
Like PMF, CDF is a complete description of random variable.
Flip the coins # of H
Properties of CDF
"starts at 0 and ends at 1"
- b) For all ,
"non-decreasing in x"
- c) For all
"probabilities can be found by difference of the CDF"
- d) For all ,
"CDF is right continuous"
- e) For
"For a discrete random variable, there is a jump (discontinuity) in the CDF at each value . This jump equals
- f) for all
"Between two jumps the CDF is constant"
Continuous Random Variables
outcomes uncountable many
T: arrival of a partical
For any random variable (continuous or discrete)
- b) is nondecreasing in
- d) is right continuous
where A, B are intervals of the same length contained in [0,1]
Probability Density Function (PDF)
discrete: PMF <--> CDF (sum/difference)
continuous <---> (derivative/integral)
Theorem: Properties of PDF
- a) ( is nondecreasing)
Some useful continuous Random Variables
Gaussian (Normal) R.V.