Basic Ideas in Probability and Statistics for Experimenters: Part I: Qualitative Discussion. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute [email protected] http://www.ecse.rpi.edu/Homepages/shivkuma Based in part upon slides of Prof. Raj Jain (OSU).
Rensselaer Polytechnic Institute
Based in part upon slides of Prof. Raj Jain (OSU)
He uses statistics as a drunken man uses lamp-posts –
for support rather than for illumination … A. Lang
How to make
How to avoid
•P is the Probability Mass function if it maps each event A, into a real number P(A), and:
ii.) P(S) = 1
iii.)If A and B aremutually exclusiveevents then,
…In fact for any sequence of pair-wise-mutually-exclusive events, we have
Derived by breaking up above sets into
mutually exclusive pieces and
comparing to fundamental axioms!!
Probability Distribution Function (pdf)
a.k.a. frequency histogram, p.m.f (for discrete r.v.)
Expectation (Mean) = Center of Gravity
You can drown in a river of average depth 6 inches!
for all x.
• In general, P(X = x) =
Binomial Variables are useful for proportions (of successes. Failures) for a small number of repeated experiments. For larger number (n), under certain conditions (p is small), Poisson distribution is used.
Binomial can be skewed or normal
p and n !
Where > 0 is a constant
and E(X) =
for some > 0
Gaussian/Normal Distribution: Why?
looks nothing like
bell shaped (gaussian)!
Large spread ()!
CENTRAL LIMIT TENDENCY!
Sample mean of uniform distribution (a.k.a sampling distribution), after very fewsamples looks remarkably gaussian, with decreasing !
Important Random Variables: Normal
(a.k.a. unit normal deviate)
Height & Spread of Gaussian Can Vary!
Rapidly Dropping Tail Probability!
is an excellent estimate of true mean.
be rejected with HIGH confidence!
Meaning of Confidence Interval
Statistical Inference: Is A = B ?
Step 1: Plot the samples
Compare to (external) reference distribution (if available)
Since 1.30 is at the tail of the reference distribution,
the difference between means is NOT statistically significant!
Random Sampling Assumption!
Under random sampling assumption, and the null hypothesis
of yA = yB, we can view the 20 samples from a common population
& construct a reference distributions from the samples itself !
t-distribution: Create a Reference Distribution from the Samples Itself!
Statistical Significance with Various Inference Techniques
assumption not required
from samples itself!
Normal, 2 & t-distributions: Useful for Statistical Inference