Week 1
Main subjects of this week:
• Conditional probability
• Density function of a discrete random variable
• Cumulative distribution function of a discrete random variable
• Expectation and variance of a discrete variable
• Rules for linear transformations
• Bernoulli distribution
• Binomial distribution
Conditional probability
Conditional probability: the probability of an event A, with the condition
that event B also occurs.
!(" ∩ #)
! "# =
!(#)
! " ∩ # = )ℎ+ ,-./0/121)3 )ℎ0) " 045 # /.)ℎ .667-.
Density function of a discrete random variable
Discrete random variable: has a finite number (countable) of outcomes. The
probability density function (pdf) of a discrete variable is defined by f(x) =
P(X = x) for all outcomes x of X. All outcomes are positive and the total area
under f(x) equals 1.
Cumulative distribution function of a discrete random variable
The cumulative distribution function of a discrete random variable is defined
by F(a) = P(X≤a) and is a non-decreasing graph with jumps.
,Expectation and variance of a discrete variable
9 : = ; ∗ =(;)
>
? : = (; − A)B ∗ =(;)
>
CD : = E = EB
Short-cut formula for variance
? : = 9 : B − 9(:)B
Rules for linear transformations
If Y = a + bX is a linear transformation of X, then there is also a relationship
between E(Y) and E(X) and V(Y) and V(X). The rules are as follows:
Mean E(Y)
AF = 0 + /A>
Variance V(Y)
EHB = / B E>B
Bernoulli distribution
A variable that can only have two outcomes (0 or 1). Where 0 = failure and 1
= success.
P(X = 1) = the probability of success (p)
P(X = 0) = the probability of failure (1-p)
It is denoted as :~#+-4(,)
E(X) = p
V(X) = p(1-p)
2
, Binomial distribution
A binomial experiment has the parameter p and consists of a series of
independent and identical repetitions of a Bernoulli experiment. You can see
it as multiple Bernoulli experiments performed after each other. Again, you
can only have two outcomes (success and failure) in each repetition (also
called a trial).
4 L
! J=K = , (1 − ,)NOL
K
Where:
k = the number of successes
n = number of trials
p = the probability
Week 2
Main subjects of this week:
• Density function of a continuous random variable
• Cumulative distribution function
• Expectation and variance
• Uniform distribution
• Normal distribution
Functions of a random variable
Discrete: finite or countable number of outcomes
Continuous: all values in an interval are possible outcomes
Probability density function (pdf): the probability that X = x
Cumulative distribution function (cdf): the probability that X ≤ a
3
Main subjects of this week:
• Conditional probability
• Density function of a discrete random variable
• Cumulative distribution function of a discrete random variable
• Expectation and variance of a discrete variable
• Rules for linear transformations
• Bernoulli distribution
• Binomial distribution
Conditional probability
Conditional probability: the probability of an event A, with the condition
that event B also occurs.
!(" ∩ #)
! "# =
!(#)
! " ∩ # = )ℎ+ ,-./0/121)3 )ℎ0) " 045 # /.)ℎ .667-.
Density function of a discrete random variable
Discrete random variable: has a finite number (countable) of outcomes. The
probability density function (pdf) of a discrete variable is defined by f(x) =
P(X = x) for all outcomes x of X. All outcomes are positive and the total area
under f(x) equals 1.
Cumulative distribution function of a discrete random variable
The cumulative distribution function of a discrete random variable is defined
by F(a) = P(X≤a) and is a non-decreasing graph with jumps.
,Expectation and variance of a discrete variable
9 : = ; ∗ =(;)
>
? : = (; − A)B ∗ =(;)
>
CD : = E = EB
Short-cut formula for variance
? : = 9 : B − 9(:)B
Rules for linear transformations
If Y = a + bX is a linear transformation of X, then there is also a relationship
between E(Y) and E(X) and V(Y) and V(X). The rules are as follows:
Mean E(Y)
AF = 0 + /A>
Variance V(Y)
EHB = / B E>B
Bernoulli distribution
A variable that can only have two outcomes (0 or 1). Where 0 = failure and 1
= success.
P(X = 1) = the probability of success (p)
P(X = 0) = the probability of failure (1-p)
It is denoted as :~#+-4(,)
E(X) = p
V(X) = p(1-p)
2
, Binomial distribution
A binomial experiment has the parameter p and consists of a series of
independent and identical repetitions of a Bernoulli experiment. You can see
it as multiple Bernoulli experiments performed after each other. Again, you
can only have two outcomes (success and failure) in each repetition (also
called a trial).
4 L
! J=K = , (1 − ,)NOL
K
Where:
k = the number of successes
n = number of trials
p = the probability
Week 2
Main subjects of this week:
• Density function of a continuous random variable
• Cumulative distribution function
• Expectation and variance
• Uniform distribution
• Normal distribution
Functions of a random variable
Discrete: finite or countable number of outcomes
Continuous: all values in an interval are possible outcomes
Probability density function (pdf): the probability that X = x
Cumulative distribution function (cdf): the probability that X ≤ a
3
