1. שיטות כמותיות בחקר רשתות פרופ' רן גלעדי המח' להנדסת מערכות תקשורת
Basic Statistics Rehearsal
שיטות כמותיות בחקר רשתות
פרופ' רן גלעדי
המח' להנדסת מערכות תקשורת

2. Goal of this chapter
to go again through basics of statistics, and mainly:
Confidence level
Hypothesis testing
Significance level
t-student test
Goodness of fit
χ2 test

3. Fundamentals – or go back to 1st year
Given n random variables:
The Mean (expected value) of is or E[ ]
The variance of is or Var( ),
Standard deviation is
Suppose are IID random observations.
Sample mean is unbiased estimator of
Sample variance is unbiased estimator of
Mean
Median
Mode

4. The normal density curve
The height of a normal density curve at any point x is given by:
The standard normal distribution is the normal distribution with mean 0 and standard deviation 1, denoted as N(0,1).
The standardized value of x is defined as
which is also called a z-score (it means how far x is from the mean in terms of deviations).

5. The 68%, 95%, 99.7% rule

6. Point / interval estimation
A point / interval estimator draws inference about a population by estimating the value of an unknown parameter using interval or a point (single value): ?
Population distribution
Parameter
Sample distribution
Point estimator
Interval estimator
Statistical Inference

7. Back to basics – (or to 1st year)
by itself is a random variable, every sequence i generates
So,
But, we don’t know
However – in simulations observations are correlated, so all the above is irrelevant. S2(n) has negative bias:

8. Confidence level & interval for m
To estimate m, a sample of size n is drawn from the population, and its mean is calculated.
Under certain conditions, is normally distributed (or approximately normally distributed), thus, for large n and the central limit theorem:
This means
Or, eventually:

9. W.S. “Student” Gossett (1876 - 1937)
1905

10. Back to confidence level & interval
What if σ is unknown?
We use S2(n) instead. If n is large enough, we use
For skewed (nonsymmetrical) distributions, bigger n is required.
In case of normal distribution, tn is t-distribution with n-1 df (degrees of freedom).
For n>1, in this case, confidence level (C) and interval can be determined precisely:

11. The student t-distribution

12. t-student applications
Test that a sample Xi comes from a normally distributed population with μ, i.e., X=N(μ,σ2):
We have 10 measurements: 10.1, 10.1, 10.5, 10.7, 10.9, 11.1, 11.2, 11.2, 11.4 and Sample average is: Can it be population average of 10.5? S2 is: Checking for average of 10.5, t is From the tables of the student t-CDF for 9 df, we find that Pr(t<2.3;9)=97.65%. Since 2.35% support this average – it is unlikely.
Test difference between two samples means:
Set of 6 sticks: 11.23, 11.37, 11.04, 11.14, and are compared to set of 7 sticks: 11.38, 11.03, 11.26, 11.45, 11.45, and Assuming the two sets are N(μ,σ2) and have the same variance, are they from the same origin? X1=11.255, X2=11.306; S12=0.0257, S22=0.0241, S2=
Since we need the “two tailed test”:
Pr(|t|>0.582)=2[1-p(0.58;11)]≈2[1-0.72]=0.56, so, it is likely they are from the same origin.

13. More applications Calculate μ with some confidence level:
10 observations are from a normal distribution with unknown mean, X=N(μ,σ2): 1.20, 1.50, 1.68, 1.89, 0.95, 1.49, 1.58, 1.55, 0.50 and Sample average is 1.34, sample variance is What is the 90% confidence interval for the average?
.99
.95 .9
n-1
63.657
12.706
6.314 1
9.925
4.303
2.920 2
5.841
3.182
2.353 3
4.604
2.776
2.132 4
4.032
2.571
2.015 5
3.707
2.447
1.943 6
3.499
2.365
1.895 7
3.355
2.306
1.860 8
3.250
2.262
1.833 9
3.169
2.228
1.812 10
2.576
1.960
1.645 ∞

14. How to use student t-tables
= .05
tA, n-1
t.100
t.05
t.025
t.01
t.005

15. Concepts of Hypothesis Testing
Two hypotheses:
H0 - the null hypothesis
The statement of “no effect” or “no difference” = default
Ha - the alternative hypothesis
The statement we hope or suspect is true (when H0 is rejected)
Usually one would decide on Ha first (formulate it)
Significance level, α, is a measure of how far the deviation from the expected behavior to justify rejection. Significance level of a test is usually 1% or 5%. It also denotes the risk of Pr[Error I]
Reject
Don’t reject
Decision
Truth
Error I √ H0
Error II
not H0

16. Goodness of fit tests
The oldest GOF hypothesis test is the χ2 test from 1905, by Pearson.
χ2 test essentially compares a histogram to a fitted probability density function (pdf).
There are n observations (Xi’s), and we want to check H0 that the Xi’s are IID, random variables with some (pdf) .
Histogramming: Divide the entire range into k adjacent intervals
Count the Xi’s in each interval (= )
Calculate the expected proportion of the Xi’s in each of the interval from the fitted pdf (=nj)
Now build the test statistic χ2:
If H0 is true, χ2 should be low. How low? Tables…

17. χ2 distribution

18. χ2 applications The Prussian cavalrymen killed by horse kicks:
L. von Bortkiewicz, “Das Gesetz der Kleinen Zahlen”, Teubner, Leipzig, 1898.
Records of 20 years, from 10 army corps (200 samples)
Data:
Total number of deaths:
So, average number of deaths per year per corp:
H0: number of deaths are Poisson-distributed, i.e.,
This is a composite hypothesis, since depends on unspecified . So, estimate and calculate Npk .
Finally, calculate χ2 = 0.32, and compare to χ2 with df 2.
χ2(2,95%)=5.99, so 0.32 clearly can’t reject H0.
Total 4 3 2 1
Number of deaths, k
200 22 65
109
Number of observations of k, fk

19. Another one Results of experiment, observations,…
Calculated χ2 is 20.6
How many degrees of freedom? n-1=5
At =.05, χ2 from tables is 11.07
Or use CHIINV(5,.05)
Conclusions?

20. P-Value (measures significance level)
Get p-value of χ2 test value
df =5
χ2 =20.06
CHIDIST(20.06,5)=.00094
p=.00094
This is an alternative way of presenting the results