PMF
a)
b)
c) event
success probability
1) Flip a coin
# of H
2) Manufacture a Chip
# of acceptable chips
3) Bits you transmit successfully by a modem
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
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
Like PMF, CDF is a complete description of random variable.
Flip the coins
# of H
- a)
![{\displaystyle F_{X}(-\infty )=0\Leftarrow F_{X}(-\infty )=P[X\leq -\infty ]=0\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/155b159e6f50caab61268e7a420484115cd95577)
"starts at 0 and ends at 1"
- b) For all
, ![{\displaystyle F_{X}(x')\geq F_{X}(x)\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b82ce5d143130f25fa959ee306bcc8280a9e79c7)
"non-decreasing in x"
- c) For all
![{\displaystyle x,x'\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c43cedf9e8be3498c88a6adc321c3af81a369e90)
"probabilities can be found by difference of the CDF"
- d) For all
,
"CDF is right continuous"
- e) For
![{\displaystyle x_{i}\in S_{X}\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abcd516e761f69ca637c0e61c53e320ab8b416a2)
"For a discrete random variable, there is a jump (discontinuity) in the CDF at each value
. This jump equals
- f)
for all ![{\displaystyle x_{i}\leq x\leq x_{i+1}\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/316a1717cf171c3cd2eb4570530d627b1ac12548)
"Between two jumps the CDF is constant"
- g)
![{\displaystyle P[X>x]=1-\underbrace {F_{X}(x)} _{P[X\leq x]}\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49fcbca9f23a05f5084af91ed8177776aa16929e)
outcomes uncountable many
T: arrival of a partical
V: voltage
: angle
: distance
No PMF,
For any random variable (continuous or discrete)
- a)
![{\displaystyle F_{X}(-\infty )=0\quad F_{X}(\infty )=1\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8a7088862faf3e965e8d79153311ed2fd009bf)
- b)
is nondecreasing in ![{\displaystyle X\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee148541ea902ae79e79d4a9816cb30597867e06)
- c)
![{\displaystyle P[x<X\leq x']=F_{X}(x')-F_{X}(x)\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23bf32969db9e5ddd9bb688b2c6b407124ef9ba5)
- d)
is right continuous
where A, B are intervals of the same length contained in [0,1]
(exercise)
discrete: PMF <--> CDF (sum/difference)
continuous <---> (derivative/integral)
- a)
(
is nondecreasing)
- b)
![{\displaystyle F_{X}(x)=\int _{\infty }^{x}f_{X}(x)dx\quad (F_{X}(\infty )=0)\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d8319d2cf65b13542af01299b93cf76eb1d0df)
- c)
![{\displaystyle \int _{-\infty }^{\infty }f_{X}(x)dx=1\quad (F_{x}(\infty )=1)\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92191df60a4b9cfa24ced85b2e9924859a329a00)