discret_distributionssssssssssssssssss.pdf

discrete series

1. Probability Distributions
Vikash R. Satyal
Today:-
Binomial distribution
Poisson distribution
Binomial approximation

2. What is probability distribution?

3. Can you answer?
If a fair coin is tossed 5 times what is the
probability of getting Head 2 times?
Ans0.3125,
Binomial distribution

4. Random variables( RV )
Variables that are generated from some random
process, (not any variable as in math)
Basic element of probability distribution
Helps to simplify probability problems
Example: Random Variable
•Sample of household from a community gives random
income levels
•Covidinfected peoples who enter Tekuhospital
•# girl reaching bus stop between 10:11 AM

5. Discrete & Continuous RV
Discrete rv–“count values”
•Number of Dengue infected people
•Number of bad checks received by a restaurant
•Number of absent employees on a given day
Continuous rv–“values in an interval
Life hour of a electrical bulb from a factory
Arrival time of customers or queue at a bank
Percent of the labor force that is unemployed

7. Some Special Distributions
•Discrete
–Binomial
–Poisson
–Hypergeometric
•Continuous
–Uniform
–Normal
–Exponential
–t
–Chi-square
–F

9. Feature of Binomial Distribution
Discrete rv
Each trial (toss) has only two possible outcomes:
heads or tails, yes or no, success or failure.
The probability of the outcome of any trial (toss)
remains fixed over time. With a fair coin, the
probability of heads remains 0.5 each toss
regardless of the number of times the coin is
tossed.
The trials are statistically independent
Do we find Binomial situations in life?

10. P(2 Heads in 10 toss of a coin) = P(X=2)
n!= n x (n-1) x (n-2) x ………3x2x1
5! = 5x4x3x2x1=120
p =success(Headin a toss),
q =not success(Tailin a toss),

11. Binomial problem solving
If a fair coin is tossed 10 times what is the
probability of getting Head 2 times?
Let X= # time we get HEAD
‘n =10 toss, p = 0.5 = q (=1-p)
Mean of X for this case: E(X)=np=10x0.5=5
Stdof X=npq=10x0.5x0.5=2.5

12. Your turn1
Suppose the exit poll of some election at randomly selected booths
indicates that the probability of the voters with Party ABC is 0.55,
and the probability that they are with Party PQR is 0.30.
Assuming that these probabilities are accurate, answer the following
questions pertaining to a randomly chosen group of 10 Voters.
(a) What is the probability that four are with Party PQR?
(b) What is the probability that none are with Party ABC?
(c)What is the probability that at least eight are with Party PQR?

13. Coin tossing, additional issues
If a fair coin is tossed 10 times what is the probability of
getting,
a)at least 3 heads? b)at most 1 head c)no head
Let X= # getting head
‘n =10 toss, p = 0.5 = q
b) P(X≤ 1)=P(X= 0) +P(X= 1)
c) P(X=0)

14. Using EXCEL to solve
Binomial problems
=P(X=2)=BINOM.DIST(2,10,0.5,FALSE)=0.0439
If a fair coin is tossed 10 times what is the probability of
getting, a)at most 1 head b)at least 3 heads? c)no head
a)P(X≤ 1)=BINOM.DIST(1,10,0.5,TRUE) = 0.0107
b)P(X≥ 3) =1-P(X≤ 2)
=1-BINOM.DIST(2,10,0.5,TRUE) =0.945
c)P(X=0) = BINOM.DIST(0,10,0.5,FALSE) =0.000977

15. Mean and variance of Binomial Distribution
Example: A packaging machine produces 20 percent defective packages. A
quality controller took a random sample of 10 packages. Assuming that the
defective packages follow Binomial distribution, what is the expected
number of defective packages he will get and what will be its standard
deviation?
Solution: Let X=#defective packages. Here as
p=Prob(defective)=0.2,
so q=Prob(non defectives)=0.8
and n=10.

16. Let X= #votes. We have p=.55, q=.45, n=160
•Mean= np =160x0.55=88 votes in favour
•Variance = npq= 160x0.55x0.45=39.6 votes
Your turn2
In the coming local election, exit poll of randomly selected voters
indicates that the probability that Nepali voters are with Party
ABC is 0.55. In a small village where there are only 160 voters
what expected votes party ABC should expect and what will be
the variance of the vote?

17. break

18. Poisson Distribution
(Poisson=French pronunciation: ​[pwasɔ̃]
Siméon-Denis Poisson |
French mathematician |

19. Poisson Distribution
Discrete rvwithn independent & identical trials
#trials very large(n∞)
p=prob. for favorable event is very small(p
0)(Rare events)
Expected # occurrence is constant for an
experiment
Henc, nxp=λ=constant (called Lamda)

20. What events can be considered Poisson?
•Defects in manufactured goods
–number of defects per 1,000 from a factory
–number of accidents in a busy street
–number of errors per typed page
•Arrivals at queuing systems
–airports --people, airplanes, automobiles,
baggage
–banks --people, automobiles, loan applications

21. Poisson Distribution
Mean = 
Variance = 
Std=√λ
Probability function

22. Example:Suppose that we are investigating the safety of a dangerous
intersection. Past police records indicate a mean of five accidents per
month at this intersection. The number of accidents is distributed
according to a Poisson distribution, and the Highway Safety Division
wants us to calculate the probability in any month of exactly 0, 1, 2
accidents.

23. Example:Suppose that we are investigating the safety of a dangerous intersection. Past police records
indicate a mean of five accidents per month at this intersection. The number of accidents is distributed
according to a Poisson distribution, and the Highway Safety Division wants us to calculate the
probability in any month of exactly 0, 1, <7, >8 accidents.
Solution using EXCEL
=POISSON(X, λ, Cumulative(TRUE/False))
Here λ=5, & for X=0
a)P(X=0) we use, =POISSON(0,5, False) =0.0067
b)P(X=1) we use, =POISSON(1,5, False) =0.034
c)P(less than 7) =P(X<=6) we use, =POISSON(6, 5, True)=0.762
d) P(more than 8) =P(X>8) = 1-P(X<=7)
=1-POISSON(7, 5, True) =0.133

24. Your turn3: Find the probabilities
•The number of work-related injuries per month in a
manufacturing plant is rare but happens with a mean of 2.5
a month.
•What is the probability that in a given month
a)no work-related injuries occur?
b) at least one work-related injury occurs?

25. Poisson Distribution as an Approximation of the
Binomial Distribution
If for a Binomial process(random variable), we have
sample size >20 & probability of happening(p) ≤ 0.05
we can use Poisson formula(easier) to calculate probability
Ex: In a hospital with 20 kidney
dialysis machines chance of any one
of them malfunctioning during any
day is 0.02. What is the probability
that exactly three machines will be out
of service on the same day?

26. Thanks