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Discrete probability distribution, Normal distribution , &
Sampling distribution
•Probability is a numerical measure of the like hood that an event will occurs
•Simple event is a outcome from one simple space and joint event is event that occurs
simultaneously
...
P(A∩B)
= P(B)P(A|B)
...
+P(A|Bi)P(Bi)

Probability
Distribution

Discrete
Probability
Distribution

Continuous
Probability
Distribution

Binomial

Nominal

1

Probability Distribution
•The probability distribution for a
variable assigns a probability to every possible
value outcome of the variable or probability of the
value falling within a particular interval
Study were conducted on bus operation in the university
...
Probabilities for the number of student waiting at the main bus stop were
computed during any time period
...
Mean arrival rate is 2 students per minutes
...


X

Probability

0
1
2
3
4
>5

0
...
1804
0
...
3210
0
...
1024

Discrete probability distribution
•Assumes each of its values or number with a certain probability
...
g
...

0 = male and
1=female
0,1,2,…
...

1
...
Expected value of probability distribution
3
...
g
...
Number of ways books could be arranged is 6P4= 6!/2! = 360
•Rule 4: combination (ordered doesn’t matter)
nCx= n!/(x!(n-x)!)
how many combination would there be for E,R,B when two letters are taken at a time 3C2 = 3!/(3!x1!) = 3 i
...
ER,
EB, RB
•But with repetition, there would be nCx= (n+x-1)!/(x!(n-x)!)

Mean & variance
•Mean (μ)
μ = np , p is probability of success for each trial
•Variance (σ2)
σ2 = npq

Graphing
•The graph of a binomial distribution can be constructed by using all the possible X values of a
distribution and their associated probabilities
•The X values usually are graphed along the X axis and the probabilities are graphed along the Y
axis

4

e
...
: probabilities for three binomial distribution with n=8

P(x) = [n!/(x!(n-x)!)][pxqx] P(0) = [8!/(0!(8-0)!)][0
...
80] = 0
...
The shape of binomial
distribution a
...
5 b
...
5 c
...
5
•as n increases, the distribution of x becomes more symmetrical regardless the value of p
...

Holding on the assumption of independency, analysis of data shows average number of cars
arriving in 15 minute period is 10
...


Hypergeometric distribution
•discrete probability distribution where the random variable is the number of successes in a
sample containing observations from a finite population without replacement

•trials are not independent, success and failure change from trial to trial

•in binomial distribution sample data are selected: with replacement from a finite population
or without replacement from an infinite population – implying probability of success p is
constant for all observations)
6

•Hypergeometric probability function

p(x) =

, for 0
example
Light bulbs produced by Phillips international were packed in boxes of 12 units each
...
If the defective have 5 defect bulbs, what the probability the
inspector will find one of the three fuses defective
...
discrete probability distribution
–the probability that a continuous random variable will assume a particular value is zero
–cannot be expressed in tabular form
–described by an equation or formula called a probability density function or sometimes referred to
as a density function
Probability density function
•properties:
–since the continuous random variable is defined over a continuous range of values (domain of
the variable), the graph of the density function will also be continuous over that range
–the area bounded by the curve of the density function and the x-axis is equal to 1, when
computed over the domain of the variable
–the probability that a random variable assumes a value between a and b is equal to the area
under the density function bounded by a and b
...
It is symmetrical
thus the mean equals the median

–Exponential distribution is a right-skewed (mean > median) and ranges from zero to positive
infinity

–The normal distribution bell-shaped, symmetrical (mean = median = mode), where the location is
determined by the mean and its spread is determined by the standard deviation
...
718
•π = the mathematical constant approximated by 3
...
Assume we repeatedly take samples of a given size from
population and calculate sample mean for each sample
...
The distribution of these mean is the
sampling distribution of the sample mean
...

The formula
•Mean of sample mean is the population mean
μx = μ
•Standard deviation of the sample means is the s
...
r of sample
size
σx=σ
√n

9

Z value for sampling distribution of mean
•Recall that Z = (x-μ) / σ
•If sample mean are normally distributed, the Z score formula applied for sample means would
be:
Z = (x- μx) / σx
= (x - μ ) / [σ / √n]

Sampling distribution of a proportion
•Sometimes, a researcher use the sample proportion to analyze a sample
...
g
...
s of sample proportion
is √(pq/n)

•z = p-p/[√(pq/n)]; p = sample proportion, p population proportion, n=sample size, p+q=1

10



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Title: Discrete probability distribution, Normal distribution , & Sampling distribution
Description: Discrete probability distribution, Normal distribution , & Sampling distribution

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