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2. Probability Distributions
2.1. Random Variables, Expected Values and Variance
Definition (random variable): A random variable is a numerical description of the outcome of an experiment.
There are two common types of random variables. They are:
Discrete random variable: a quantity assumes either a finite number of values or an infinite sequence of values, such as 0, 1, 2,
Continuous random variable: a quantity assumes any numerical value in an interval or collection of intervals, such as time, weight, distance, and temperature.
Definition (probability distribution (or density) function (p.d.f)): a function describes how probabilities are distributed over the values of the random variable.
Required conditions for a discrete probability distribution function:
Let be all the possible values of the discrete random variable X. Then, the required conditions for to be the discrete probability distribution for X are
(a)
(b)
Required conditions for a continuous probability density:
Let the continuous random variable Z taking values in subsets of . Then, the required conditions for to be the continuous probability density function for Z are
(a)
(b)
Note: As X is discrete,
.
As X is continuous,
Expected Value and Variance:
As X is discrete,
and
As X is continuous,
and
Important Properties of Expected Value and Variance:
are constants and X is discrete or continuous.
1..
2. .
3.
Example 1:
The probability distribution functions (discrete random variable) or probability density functions (continuous random variable) for a random variable X are
(a)
(b)
(c)
Find .
[solution:]
(a)
(b)
(c)
Example 2:
The probability distribution function for a discrete random variable X is
where k is some constant. Please find
(a) k (b) (c) and (d)
(e)
[solution:]
(a)
.
(b) .
(c)
and
(d)
(e) .
Example 3:
Let X be a discrete random variable representing the number of hours
a college student spending on reading novels per week. The following
probability distribution has been proposed.
, ,
where k is some constant.
(a) Compute c. (b) Compute and .
(c) Compute a nd .
[solution:]
(a)
.
(b).and
(c)
and
Example 4:
The probability density function for a continuous random variable X is
where a, b are some constants. Please find
(a) a, b if (b) .
[solution:]
(a)
and
Solve for the two equations, we have .
(b)
Thus,
Example 5:
The probability density function for a continuous random variable X is
Please find (a) (b) (c) and
(d) (e) .
[solution:]
(a)
(b)
(c)
.
Since
,
.
(d) .
(e)
Definition (cumulative distribution function (c.d.f)): .
Important Properties of CDF:
1. CDF is right continuous.
2. As X is discrete, ;
As X is continuous and is continuous, .
Example 2 (continuous):
The probability distribution function for a discrete random variable X is
Thus, the CDF is
Example 1 (a):
The probability distribution functions (discrete random variable) or probability density functions (continuous random variable) for a random variable X are
Thus, as ,
as ,
.
Thus, the CDF is
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