Summary of Analysis of Production Systems

Summary Analysis of production Systems
[4DC10]

February 11, 2021


Probability models
Probabilities
If two events are independent of each other

P (EF ) = P (E) ∗ P (F )

Conditional probability
P (EF )
P (E|F ) =
P (F )

Expected value
Expected value(Discrete) (centre of probability mass):
∞
X
E[X] = xj ∗ p j
j=1

Sum of expectation is expectation of sum

E[X + Y ] = E[X] + E[Y ]

Rule for constants
E[aX + b] = a ∗ E[X] + b
If two events are independent
E[XY ] = E[X]E[Y ]

Variance
Variance:
V ar[X] = E[X 2 ] − (E[X])2
Standard deviation: p
σ(X) = V ar[X]
Rule for constants
V ar[aX + b] = a2 V [X]
If two events are independent

V ar[X + Y ] = V ar[X] + V ar[Y ]




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, Continuous random variable
Z b
F (X) = P (a < X < b) = f (y)dy
a
Z ∞
E[X] = x ∗ f (x)dx
−∞

Uniform random variable X on [a, b] has density:
1
f (x) = ,a < x < b
b−a
x−a
P (X ≤ x) = ,a < x < b
b−a
a+b
E[X] =
2
1
V ar[X] = (b − a)2
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Density
d
f (x) =P (X ≤ x)
dx
Exponential random variable X with parameter(or rate) λ has density

f (x) = λe−λx , x > 0

P (X ≤ x) = 1 − e−λx
1
E[X] =
λ
1
V ar[X] = 2
λ

Binomial
A Binomial random variable X is the number of successes in n independent Bernoulli trials
X1 , ... , Xn , each with probability p of success.
 
n
P (X = k) = ∗ pk (1 − p)n−k k = 0, 1, ...n
k




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