Probability theory is one theory that learning problem from events that cannot predict first. Something as sample is telephone traffic, so that in telephone traffic engineering analysis and probability theory is a mathematic model of observed symptom.
Application of probability theory on traffic engineering, do with experiments (observation and data collection) and analyzing the result. Simple sample about it:
Experiments:
Observing telephone call between nine o’clock until ten o’clock
Sample space:
If
t = call time
Then:
S = {t, 9 < style="font-weight: bold;">Probability Standard
On calculation, usually used equally likely method and relative frequency, these both are basic method of probability theory.
- Equally likely method, In this method, all sign have possibility to show that same.
- Relative frequency method, Possibility unit standard show one sign connected with how often that sign show in several experiment. There is inclination, that relative frequency on a range fixed values, so that, there is long term stability relative frequency than that fixed sign.
Several Theorems
- Addition
- Multiplication
- Bayes Theorem, this Bayes theorem giving one way to calculate, conditional probability from one sign to the other sign with calculating conditional probability from that the other sign to first sign.
Example:
Probabilities x line occupied and the other line (n-x) are free?
1
2
.
.
.
n
Every line:
Occupied Probability = p
Free Probability = q = 1 – p
Probabilities x line occupied and the other line (n-x) are free = px. q(n – x).
Total possibility occupied line x and the other free line (n-x): n ! / x ! (n-x) !
(That each have probability = px. q(n – x))
So, for occupied line x and the other free line (n-x):
P(x,n) = {n ! / x ! (n-x) !}. px. q(n – x)
This is Bernoulli distribution equation
What is meaning probability (line) occupied (or free)?
On telephone system, for every line:
p = occupied line probability, (several hour occupied in 1 hour)
q = Free Probability, (several hour free in 1 hour)
Observation with relative frequency:
| |
®| Dt |¬ ® t
Experiment: line observation every 1 hour, outcomes is:
occupied or
free
There is n = 1 (hour) / t (hour) time observation
Dt ® 0, then n ® -
Appearance occupied sign frequency = f(p)
Appearance occupied sign relative frequency = f(p)/n
= f(p). Dt = p (p + q = 1)
(The long time occupied line in hour per hour (busy))
so:
P(x, n) = æ n ö px. q(n – x)
è x ø
Several hour condition which occupied line x and the other line free (n-x), go on in certain time interval.
Random Variable and Probability Distribution
Random Variable, a quantity that the value formed by sign (outcomes) from one experiment.
Continue:
The long time occupied from one equipment (line) telephone is continuous random variable.
Probability distribution, random variable value that connected with the probability is shaping probability distribution.
Probability Function:
P( X = x ), where X is random variable and x a value that taken.
Continue:
Probability density function (p.d.f) and function distribution (d.f) or Probability Distribution Function (P.D.F)
Random variable T :
o p.d.f = f(t)
o d.f = P.D.F = F(t)
t
F(t) = P(T £ t) = ∫ f(u).du or f(t) = d.F(t) / dt
p.d.f : a standard from a density (closeness/concentration/density) from one probability on every point in continuous random variable time interval.
