Download Working with Probability ~ 2

Working with Probability ~ 2
Joseph Spring
School of Computer Science
Measures and Models of Software Engineering
Areas for Discussion
• Random Variables
• Probability Distributions
• Probability Density Function
• E(X) and Var(X)
Random Variables
There are two types of random variable:
– Discrete
– Continuous
A random variable (also referred to as variate)
is a function associating events/outcomes
from a sample space to numeric values,
subject to chance (i.e. probability)
Random Variables
For Example
• The number of threes resulting when 2 dice
are thrown
• Amount of pocket money received given that
it is a multiple of the value obtained when a
die is thrown
• Winnings from the throw of a die / turn of a
roulette wheel / turn of a card / …
Random Variables
• Random Variables (r.v.) are represented by
capital letters (X, Y, Z, T, etc; …)
• Particular values that a r.v. can take are
represented by lower case letters (x1, x2, .... )
• the function P that allocates probabilities is
referred to as the pdf - ( the probability
density function )
Properties of a Random Variable
• Discrete r.v.
– variable takes only discrete values (no continuous
values)
– variable takes a finite number of possible values
x1, x2, .... , xn
– Σpi = 1 where pi is P( X = xi )
(P is the pdf)
So if you add the associated probabilities you get 1
Examples of Discrete Random Variables
• Uniform Distribution
• Binomial Distribution
• Poisson Distribution
• ….
Properties of a Random Variable
• Continuous r.v. (height, mass, time, …)
– variable takes only continuous values
– variable defined over a range of values a ! x ! b
so an infinite number of possible values for a ≠ b
– the pdf is now represented by a function f(x) and
probabilities are determined by the area under the
curve given by f(x), (as with Normal Distribution)
Examples of Continuous Random Variables
• Uniform Distribution
• Exponential Distribution
• ….
• Normal Distribution
Properties of a Random Variable
• Drawing from lecture
Examples - Discrete
1 Trivial case
For a die we could take:
X = { 1, 2, 3, 4, 5, 6 }
the number of dots on each face
The associated pdf would be:
P( X = xi ) = 1/6
which holds for all values of x
Examples - Discrete
This gives the following probability distribution:
X
= x
P ( X
i
= x
i
)
1
2
3
4
5
6
1 /6
1 /6
1 /6
1 /6
1 /6
1 /6
Examples - Discrete
2 A tetrahedral die has numbers 1, 2, 3, 4 on its faces.
The die is biased so that the probability of the die
landing on the number n is inversely proportional to n
So for example P( X = 3 ) = k/3 where k is a constant.
Given that X is a r.v. representing the number that the
die lands upon after one throw of the die find the
probability distribution for X and solve for k.
Examples - Discrete
Solution:
X
= x
i
P ( X = x i)
1
2
3
4
k /1
k /2
k /3
k /4
X is a r.v.
So k/1 + k/2 + k/3 + k/4 = 1
Hence (25/12)k = 1 and therefore k = 12/25
Examples - Discrete
The resulting probability distribution is:
X
= x
P (X
i
= x i)
1
1 2 /2 5
2
6 /2 5
3
4 /2 5
4
3 /2 5
E(X) and Var(X)
E(X) and Var(X) are now the symbols we use
to denote
– the expectation/expected value of the r.v. X
– and variance (respectively) of the r.v. X
Areas for Discussion
• Random Variables
• Probability Distributions
• Probability Density Function
• E(X) and Var(X)