Lecture 6-Probability Distributions

Question 1

Random Variables

Answer 1

In general, it is usually possible to convert points in the sample space into real values.
*The precise values cannot be known in advance since they depend on outcomes which are themselves random. *It is for this reason that variables like X are called random variables.
In more formal terms, a random variable is defined to be a function that “…assigns exactly one value to each point in a sample space for an experiment” Above, we implicitly defined our random variable to be the amount of money we receive in the game.
*The nature of the assignment gives the random variable its name. In our experiment above, X is the prize money from the game of tossing two ‘fair’ dice.
*Random Variables form the core of Inferential Statistics.

Question 2

Two types of data and two types of random variables

Answer 2

Discrete Data
Continuous Data

Discrete random variables
Continuous random variables

Question 3

Discrete random variables

Answer 3

our random variable X could assume any one of eleven possible values.
* This is an example of what is known as a discrete random variable.
*A discrete random variable is one which can take precise values on the real line. It assumes countable values, where the consecutive values are separated by a certain gap.

Question 4

Continuous random variables

Answer 4

is one which can assume any value contained in one or more intervals on the real number line.

Question 5

What is a probability distribution

Answer 5

links a random variable X with the probability that X assumes a discrete value or a range of values
*This can be presented by a table, function or formula
*Random variables can be discrete or continuous
*Probability distributions are also correspondingly discrete or continuous

Question 6

Discrete vs Continuous Random Variables

Answer 6

Every random variable has associated with it a probability distribution.
* As you may imagine, discrete random variables possess discrete probability distributions, and continuous random variables possess continuous probability distributions.
*As both types of random variables can be distinguished by their characteristics, so too can both types of probability distributions be distinguished.

Question 7

Properties of Discrete Probability Distributions

Answer 7

There are two characteristics that the probability distribution of a discrete random variable must possess
*For any value x on the real line:
(1) Pr(X=x) 0
(2) Pr(X=x) = 1

Question 8

Properties of Continuous Probability Distributions

Answer 8

Let f(x) be the probability distribution of a continuous random variable.
*Then
(1) f(x) is greater than 0 for each value of x
(2)
*The second criterion means that the area under the curve of f(x), the probability density function, is equal to 1.

Question 9

Probability Distribution for a Discrete Random Variable

Answer 9

Such a table presents all values taken on by the random variable and their corresponding probabilities.
*Furthermore, the probabilities sum to one.
*Such a table is called a probability distribution for our random variable X.

Question 10

Another example of a probability distribution for a discrete random variable

Answer 10

Consider the experiment of tossing a ‘fair’ coin twice.
* Suppose that this experiment forms part of a game during a lime among some friends.
*The rules of the game specify that each outcome of a ‘Head’ entitles the player to a $2.00 prize while each outcome of a ‘Tail’ results in the player paying out $2.00. Each time that the experiment is repeated the player stands to get net prize money of $2.00 or $0.00 or pay out $2.00. We may not, however, know the precise prize money in advance as these moneys all depend on outcomes that are also random.

Question 11

Ppro

Answer 11

Net Prize X Outcome(s) Probability
$-4 TT 0.25
$0 HT, TH 0.25 + 0.25 = 0.50
$4 HH 0.25

Question 12

Charting the Discrete Probability Distribution

Answer 12

So far we have presented probability distributions in tabular form
*It seems logical that we can also present them in graphical form, charting the values of the random variable against the probability that the random variable assumes those values
*We can proceed to show a chart of the only probability distribution among the three tables.
*We call such a chart the graph of the probability distribution.

Question 13

Example of a Discrete Cumulative Probability Distribution

Answer 13

Probability Cumulative Probability
Distribution Distribution
x P(x) x P(X < x)
1 .15 1 .15
2 .34 2 .49
3 .28 3 .77
4 .23 4 1.00

Question 14

Probability Distribution for a Continuous Random Variable

Answer 14

We define the probability distribution for continuous random variables differently.
* Recall that we cannot count the values assumed by a continuous random variable.
* The number of values taken on by the variable in any interval is infinite.
* Accordingly, we modify the approach used for the discrete random variables.

Question 15

Probability density function

Answer 15

The probability distribution function of a continuous random variable

Question 16

Probability density curve

Answer 16

The graph of this function

Question 17

The probability that the continuous random variable lies between

Answer 17

The probability that the continuous random variable lies between any two given values a and b (i.e. P(a<X<b) is given by the area under the probability density curve bounded by the lines x = a and x = b.

Question 18

Area under the Density Curve

Answer 18

Recall that the heights of the bars in the relative frequency histogram sum to 1. Hence the area under the histogram is 1.
*The relative frequency polygon has an area that approximates the area of the histogram i.e. the area under the polygon approximates to 1.
*The smoothed relative frequency polygon is the area under the probability density curve. Hence the area under the curve also equals 1.

Question 19

Probability for Continuous Random Variables

Answer 19

Hence all probabilities will lie in the range of 0 to 1 inclusive
*Also, the probability that the random variable will assume all possible intervals of values equals the entire area under the curve i.e. an area of 1.
*The axioms of probability hold.
Further the probability that the continuous random variable X assumes a single value is seen to be the area of a bar with zero width. i.e. such an area equals zero.
In other words, for continuous random variables;
P(X = a) = 0 and P(X = b) = 0

Question 20

Expected Value Of A Discrete Random Variable

Answer 20

X -4 0 4
P(X = x) 1/4 2/4 1/4

Recall the example of tossing a coin twice, and being paid $2 for H and losing $2 for T
*Suppose that we played this game repeatedly, say 4000 times.
*From the probability distribution we can say that:
–Net prize money of -$4 is expected in 25% of the games i.e. in 1000 games
–Net prize money of $0 is expected in 50% of the games i.e. in 2000 games
–Net prize money of $4 is expected in 25% of the games i.e. in 1000 games.

What then would be our average net prize money?
Avg. Net Prize Money = Total Net Prize Money over
No. of games played
= 1000 (-$4) + 2000($0) + 1000($4) over
4000
Simplifying we get the Avg. Net Prize Money to be equal to
1 (-$4) + 2($0) + 1($4) = $0
4 4 4
50

Question 21

Expected Value of a Discrete Random Variable

Answer 21

We can conclude that if we played the card game a large number of times, on average, we can expect to win nothing (and, by extension, lose nothing).
* The value of $0 so computed is called the expected value E(X) of the discrete random variable X.

In short, E(X) is called the long run average value of the random variable.
*We can calculate E(X) from a discrete probability distribution by multiplying each value of the random variable X by its corresponding probability and sum the resulting products.
E(X) = xi P(xi)
*E(X) is also seen as the mean of the discrete random variable X.
52

Question 22

Expected Value of A Continuous Random Variable

Answer 22

Recall the fundamental difference in the characterization of probability for a continuous variable i.e. it is area under the probability density curve.
E(x) is computed using Integration.
+ infinity sign
E(X) = x f(x) dx
- infinity sign
where f(x) is the probability density function of the random variable X.

Question 23

Properties of Expectation

Answer 23

Let X be any random variable. X can be discrete
or continuous.
*If a is a constant then E(a) = a
*If a is a constant then E(aX) = a E(X)
*If b is a constant then E(X + b) = E(X) + b
*If a & b are constants then E(aX + b) = a E(X) + b
If X and Y are two distinct random variables then E( X + Y) = E(X) + E(Y)
*If g(X) and h(X) are two distinct functions defined on X then
E[g(X) + h(X)] = E[g(X)] + E[ h(X)]

Question 24

Variance of Discrete Random Variables

Answer 24

E(X) has been shown to be equal to the mean of the random variable
*The Mean E(X) highlights where the probability distribution of X is centred
*There should be an associated measure of dispersion for the variable since the mean alone does not give an adequate description of the shape of the distribution.
*Unsurprisingly, variance is that measure
*It is a measure of how the values of the variable X are spread out or dispersed from the mean

Consider a discrete random variable X taking on values x2 , x2 , x3 , …… xn with associated probabilities p2 , p2 ,p3 , …… pn respectively.
* Variance can be rewritten as
P2 (x2 - )2 + P2 (x2 - )2 + …+ Pn (xn - )2
which simplifies to (weird e sign) Pi (xi - )2.
*Standard Deviation of X is the square root of the Variance of X.

Question 25

Properties of Variance

Answer 25

Let X be any random variable. X can be discrete
or continuous.
*If a is a constant then Var(a) = 0.
*If a is a constant then Var(aX) = a2 Var(X).
*If b is a constant then Var(X + b) = Var(X).
*If a and b are constants, Var(aX + b) = a2 Var(X)
*If X and Y are two independent random variables then Var( X + Y) = Var(X) + Var(Y).

Question 26

Two special discrete distributions

Answer 26

Binomial Distribution and the Poisson Distribution

Question 27

Two special continuous distributions

Answer 27

The normal distribution