Download Chapter 4: Probability models – discrete variables

Applied Statistics
Chapter 5: Probability models
1. Random variables:
a) Idea.
b) Discrete and continuous variables.
c) The probability function (density) and the distribution function.
d) Mean and variance of a random variable.
2. Probability models:
a) Bernoulli.
b) Geometric.
c) Binomial.
d) Normal.
e) Normal approximation to the binomial distribution.
Recommended reading:

Capítulos 22 y 23 del libro de Peña y Romo (1997)
Applied Statistics
5.1: Random variables
• A function which places a numerical value on each possible
result of an experiment is called a random variable.
• We use capital letters, e.g. X, Y, Z, to represent random
variables and lower case letters, x, y, z, to represent particular
values of these variables.
Discrete random variables can only take a discrete set of possible
values.
Continuous random variables can take an infinite number of
values within some continuous range.
Applied Statistics
The probability function for a discrete r.v.: is the function which
associates the probability P(X=x) to each possible value x.

The possible values of a discrete r.v. X and their respective
probabilities are often displayed in a probability distribution table:
X
x1
x2
...
xn
P(X=xi)
p1
p2
...
pn
Every probability function satisfies
p1  p2  p3 
 pn  1
The distribution function for a discrete r.v.: Let X be a r.v. The distribution
function of X is the function which gives, for each x, the cumulative
probability up to x, that is,
F ( x )  P( X  x )
Applied Statistics
Mean, variance and standard deviation of a discrete r.v.
The mean or expectation of a discrete r.v., X, which takes values x1, ,x2,
....with probabilities p1, p2,... Is given by the following expression:
  i xi P( X  xi )  i xi pi
The variance is defined by the formula
which can be calculated using
 2   i xi2 pi   2
 2   i ( xi   )2 pi
The standard deviation is the root of the variance.
Example: The probability distribution of the r.v. X is given in the following table:
xi
1
2
3
4
5
pi
0.1
0.3
?
0.2
0.3
What is P(X=3)?
Calculate the mean and variance.
Applied Statistics
5.2: Probability models
Discrete models
Continuous models
Bernoulli trials and the geometric and The normal distribution and related
binomial distributions
distributions
Applied Statistics
Bernoulli trials and the geometric and binomial distributions
A Bernoulli model is an experiment with the following characteristics:
• In each trial, there are only two possible results, success and failure.
• The result obtained in each trial is independent of the previous
results.
• The probability of success is constant, P(success) = p, and does not
change from one trial to the next.
Applied Statistics
The geometric distribution
Suppose we have a Bernoulli model. What is the distribution of the
number of failures, F, before the first success?
• P(F=0) = P(0 failures before the 1st success) = p
• P(F=1) = P(failure, success) = (1-p)p
• P(F=2) = P(failure, failure, success) = (1-p)2 p
• P(F=f) = P( f failures before the 1st success) = (1-p)f p for f = 0, 1, 2,
…
The distribution of F is called the geometric distribution with parameter p.
E[F] = (1-p)/p
V[X] = (1-p)/p2
Applied Statistics
The binomial distribution
Suppose we have a Bernoulli model. What is the distribution of the
number of successes, X, in n trials?
n r
P(Obtener r éxitos )  P ( X  r )    p (1  p )n r
r 
for r = 0,1,2, …, n.
The distribution of X is called the binomial distribution with parameters n
and p.
E[X] = np
V[X] = np(1-p)
Applied Statistics
EXAMPLE
Calculate the probability that in a family with 4 children, 3 of them are
boys.
EXAMPLE
The probability that a student has to repeat the year is 0,3.
• We pick a student at random. What is the probability that the first
repeater is the 3rd student we pick?
• We choose 20 students at random. What is the chance that there
are exactly 4 repeaters?
EXAMPLE
On average, 4% of the votes in an election are null. Calculate the
expected number of null votes in a town with an electorate of 1000.
Applied Statistics
Calculation with Excel
Binomial probabilities are tough to calculate “by hand” except in the simple cases of
zero or 1 successes or failures.
In Excel it is much
easier!
Applied Statistics
Example (Test)
Of all the charities in España, 30% are charities dedicated to children. If
50 Spanish charities are chosen at random how many of them are
expected to be dedicated to children?
Applied Statistics
Example (Test)
On average, one in every ten members of the CCOO union is a
delegate.
a) In interviews with CCOO members, what is the probability that the
first delegate will be the second person interviewed?
b) There are 4 CCOO members in La Chimbomba. What is the chance
that none of them are delegates?
c) In a sample of 100 CCOO members, what is the expected number of
delegates?
Applied Statistics
DENSITY FUNCTION OF A CONTINUOUS R.V.:
The density function of a continuous r.v. satisfies the following conditions:
It only takes non-negative values, f(x) ≥ 0.
The area under the curve is equal to 1.
Distribution function. As with discrete variables, the distribution function gives the
cumulative probability up to x, that is:
F ( x )  P( X  x )
The c.d.f. satisfies the conditions:
It takes the value 0 for any x below the minimum possible value of the
variable.
It takes the value 1 for all points over the maximum possible value of the
variable.
Applied Statistics
The normal or gaussian distribution
Many variables have a bell shaped density.
Examples:
• Weights of a population of the same age and sex.
• Heights of the same population.
• The grades in a course (urban myth).
To say that a continuous variable X, has a normal distribution with mean 
and standard deviation  , we write:
X ~ N(,2)
Applied Statistics
The standard normal distribution
The normal distribution with mean 0 and standard deviation 1 is called the
standard normal distribution.
There are tables which allow us to calculate the probabilities for this
distribution, N(0,1).
If we have a normal r.v., X with mean  and standard deviation  we can
convert this to a standard, N(0,1) r.v. using the transformation:
Z
X 

Applied Statistics
Examples
Let Z ~ N(0,1). Calculate the following probabilities
• P(Z < -1)
• P(Z > 1)
• P(-1,5 < Z < 2)
Calculate the 90%, 95%, 97,5% and 99% percentiles of the standard normal
distribution.
(These values are useful in the next chapter)
Let X ~ N(2,4). Calculate the following probabilities
• P(X < 0)
• P(-1 < X < 1)
Applied Statistics
Applied Statistics
Applied Statistics
Calculation with Excel
It is easier to do the calculations with
Excel…
with the standard normal …
… or directly with the original distribution.
Applied Statistics
Approximation of the binomial distribution using a normal
When n is large enough, the binomial distribution, X~B(n, p), looks like a normal
distribution,

N np, np(1  p)

The approximation is usually considered to be good if np > 5 and n(1-p) > 5.
EXAMPLE
We throw a fair coin in the air 400 times. What is the probability of getting
between 180 and 210 heads?
Applied Statistics
The exact solution using Excel for the binomial distribution is 0,833.
The estimated solution using the normal approximation is 0,819.
This can be improved with a continuity correction and then the
exact and approximate solutions are equal to 3 decimal places, but
if we have Excel, … why should we use an approximation?
We will see in the next chapter.
Applied Statistics
Example (Test)
According to the last CIS survey, the mean level of satisfaction with Mariano
Rajoy is 3,09 with standard deviation 2,5. If these evaluations follow a normal
distribution and a person is chosen at random, then the probability that they give
Rajoy a rating of less than 3,09 is:
a)
b)
c)
d)
0,5.
0.
1.236
1.
Applied Statistics
Example (Test)
The inflation rate follows a normal distribution with mean 1 and variance 4. Which of
the following Excel formulas gives the probability that inflation will be negative?
Applied Statistics
Example: (Exam)
The following table records the ratings of the government ministers in the last CIS
survey:
a) Supposing that the ratings of Chacón in
Spain follow a normal distribution with
mean 4,55 and standard deviation 2,6,
calculate the probability that a Spaniard
rates her below 4,55.
b) For a set of three Spanish people, what
is they probability that they all rate her
below 5?*
c) The lowest mean rated minister is
González Sinde. If her ratings are
normally distributed, calculate the
probability that a randomly chosen
Spaniard rates her exactly 5.
*See the next slide.
Applied Statistics