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Summary STA1000 Modules 3 and 4
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A comprehensive summary of modules 3 and 4 for STA1000 including additional notes.
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Module 3: Random VariablesWork Unit 1: Probability Mass and Density Functions
Random Experiment
- A procedure whose outcome (result) in a particular performance (trial) cannot be predetermined
Random Variable
- Used to capture the outcome of a random experiment (essentially the result). Often represented
by X.
X ={ X|x=1,2,3,4,5,6 } = Set notation of random variables for a die throw
1
Pr ( X=x )= for x=1,2,3,4,5∨6
6
X = Random variable. It is a variable as it can take on multiple values. It is random as it relies on
the outcome of a random event.
x = A particular outcome on any throw of the die.
Discrete Random Variable
- the set of values that can be taken on are finite or countably infinite (isolated values along the
real line).
- set notation: X ={ x| x=a , b , c … }
- takes on isolated values along the real line, usually integers. Use a PMF.
Probability Mass Function (adopted as p(x))
- defined for all values of x but non zero at a finite (or countably infinite) subset of these values.
0 ≤ p (x)≤1 for all values of x
∑ p ( x )=1 for all values of x
Continuous Random Variable
- the values that can possibly be taken on, form a continuous set.
- set notation: X ={ x|a< x <b }
- the probability of the random variable taking on an exact value is zero.
- can be measured to any degree of accuracy, usually an interval of the real line. Use a PDF!
d
Pr ( c< x <d ) =∫ f ( x ) . dx
c
Probability Density Function (adopted as f(x))
- defined for all values of x.
- 0 ≤ f ( x )≤ ∞ for all values of x.
, ∞
∫ f ( x ) . dx=1
−∞
b
Pr ( a≤ x ≤ b )=∑ p ( x )
x=a
b −1
Pr ( a< x< b )= ∑ p( x)
x=a+1
a
Sum to infinity =
1−r
term 2
a = base and r =
term 1
1
∞ x
1 2
eg. ∑ =
x=1 2 1
1−
2
Integration Concepts
- ∫ kdx=kx +c
- e.g. ∫ 4 dx=4 x +c
n+1
x
- ∫ ( x n ) dx= n+1 +c
x3 1 3
- e.g. ∫ x 2 dx= +c= x
3 3
1
- ∫ x dx=ln ( x ) +c
1
- e.g. ∫ dx =ln ( 4 ) +c
4
- ∫ cf ( x )=c ∫ f ( x ) dx (for any constant c)
x2
- e.g. ∫ 2 x dx=2∫ x dx=2 ( )
2
=x 2 +c
ax
- ∫ ax dx= ln ( a)
+c
- ∫ ( f ( x ) ± g ( x ) ) dx=∫ f ( x ) dx ±∫ g (x) dx This rule only
applies to sums and
- e.g. ∫ ( e x + 3 x 2 ) dx=∫ e x dx +3∫ x2 dx
differences!
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