Probability Distributions

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Probability Distributions
(Session 04)

Learning Objectives At the end of this session you will be able to:
solve basic problems concerning real-valued probability distributions. distinguish between discrete and continuous random variables (r.v.’s). explain what is meant by a probability distribution. calculate the population mean and variance of a given distribution.

Session Contents In this session you will
be introduced to the theory of probability distributions. be shown how to build a firm foundation of the theory of probability distributions in preparation for applications in statistical inference (Module H2). strengthen the mathematical skills that are required to deal correctly with probability ideas.

Random variables In the previous two sessions we dealt with probabilities of events. In practice events of interest are those generated by random variables. A random variable is a variable that associates outcomes in the sample space with numerical values.

An example – birth of a baby
Girl Line showing numerical scale X Boy Sample space The figure above depicts a random variable X defined as X = 0 if outcome is a boy X = 1 if outcome is a girl

A second example: Often the outcomes are actual measurements.
Thus, we could have: a random variable Y which records measurements of weights (of say maize cobs) into numbers with kilograms as units. Outcomes of any experiment can be recorded as real numbers by defining an appropriate random variable. We do this because it is easier to work with numbers.

Types of random variables
A random variable is said to be discrete if the set of possible values is countable. Examples of discrete random variables are those that records events on gender, family size, number of traffic offenses, ... A random variable is said to be continuous if the set of possible values is not countable. Examples of continuous random variables are those that record events such as weight, height, time, etc ...

Continuous to discrete?
Continuous random variables can be mapped into discrete random variables by grouping. For example, age X is a continuous random variable since it is a measure of time since birth. We can define a discrete random variable Y as Y = 1 if 0≤X<5. = 2 if 5≤X<10 = 3 if 10≤X<15 = etc. You cannot convert a discrete random variable into a continuous one.

Probability distributions
A probability distribution is a table, a function or a graph that presents possible outcomes of a trial, say E (e.g. throw of a die), together with their corresponding probabilities. Note that the outcome probabilities must sum to 1 since occurrence of E results in exactly one outcome.

Values (x) of a random variable X
An example The following is an example of a probability distribution for the gender of a new born child: Outcome Values (x) of a random variable X P(x) Male 0.5 Female 1 Total

Probability mass/density function
A probability distribution can sometimes be specified using a function f called a probability (mass/density) function. The function f must satisfy the following conditions:

Points to note: The function P(x) of the slide 10 is a probability mass function since it satisfies the two conditions above. Point 1 of slide 11 satisfies the first law of probability, as it must since P(x) represents a probability. Point 2 of slide 11 indicates that the sum is used if the set of values x is countable; otherwise the integral applies.

Expected values The weighted “centre” of a probability distribution is called the expected value written E(X). More formally the expected value of a random variable X is defined as: in the discrete case. in the continuous case. E(X) is also called the population mean and is usually denoted by  .

Example (i) If f(x) is given by then E(X) = 0(0.5) + 1(0.5) = 0.5 x
f(x) = Prob(x) 0.5 1 Total

Example (ii) Let f(x) = 2x, for 0  x  1

Moments The k-th moment of a random variable X is defined as:
in the discrete case. in the continuous case. The moments of a distribution characterize the shape of a distribution. The notation k is often used to denote the k-th moment.

Class exercise Suppose a coin is tossed twice.
Write down the possible values for the random variable X defined as: X = number of heads that occur Prepare a table showing the probability distribution function of X Use this table to determine the expected value of X

Measures of spread The variance of a random variable X is defined as
Notice that E(X2) is the second moment of X. The variance of X is also called the population variance and is denoted by 2. The square root of the variance is called the standard deviation of X. It is denoted by .

Patterns for differing variances
Note that the bigger the variance, the larger is the spread.

Skewness and kurtosis If the probability distribution is not symmetrical about the mean it is said to be skew. The distribution has a positive skewness if the tail of high values is longer than the tail of low values, and negative skewness if the reverse is true. Kurtosis is a measure of the peakness of a probability distribution. It is usually used as a comparison with the normal distribution (see later sessions) since a kurtosis of more than 3 indicates that the distribution has a higher peak than the normal distribution.

Cumulative probability distribution
In many applications we want to calculate probabilities of the type P(X≤k) or P(X>k) instead of P(X=k). The probabilities P(X≤k) for k = 0, 1, 2, .. provide an example of what is called the cumulative distribution of a random variable X. Here, the random variable X is discrete.

Some results P(X>k) = 1 – P(X ≤k). This is a direct result of the probability result that P(Ac) = 1 – P(A). Similar results can be obtained for continuous random variables. That is, if a < b then the event {X ≤ a} is a sub-event of the event {X ≤ b}. Hence P(X ≤ a) < P(X ≤ b).

Definition of F(x) The cumulative distribution at x, denoted F(x), is formally defined as: in the discrete case for a positive random variable. in the continuous case By definition, cumulative distribution is an increasing function having certain properties. These are shown below.

Results concerning F(x)
F (+ ) = 1. This says that the total area under the probability density function is 1. F(a) < F(b) for a<b. Thus F is an increasing function. P( a < X ≤ b) = F(b) - F(a). P(X = x) = 0, for every point x if X is a continuous random variable.

An example using F(x) - discrete
A discrete r.v. X, representing the number of girls in families with 5 children, has the foll: distn: X = No. of girls P(X=x) F(x) 1 2 3 4 5 Complete the table with values of F(x) What is the probability of 4 children or less?

An example using F(x) - continuous
A continuous random variable r.v. X, has probability density function given by What is its cumulative distribution function? Answer:

Practical work follows to ensure learning objectives are achieved…