The document discusses statistics and probability. It defines key concepts like random variables, discrete and continuous random variables, and probability distributions. It provides examples of discrete random variables like the number of heads in a coin toss. Continuous random variables are defined as those that can take any value, like the speed of a train. The document also gives examples of identifying discrete and continuous random variables and calculating probabilities of random variable outcomes.

1. STATISTICS AND PROBABILITY
Objective: Illustrates a random variables
(discrete and continuous).

2. Pre-Test
A.From the given random variables, find out
which ones are discrete and which are
continuous.
a) Speed of train
b) Number of students getting A grade
c) Height of men in Alaska
d) Error in measurement

3. STATISTICS
• Statistics is a branch of mathematics dealing with
the collection, analysis, interpretation,
presentation, and organization of data.[1][2] In
applying statistics to, e.g., a scientific, industrial,
or social problem, it is conventional to begin with
a statistical population or a statistical
model process to be studied. Populations can be
diverse topics such as "all people living in a
country" or "every atom composing a crystal."
Statistics deals with all aspects of data including
the planning of data collection in terms of the
design of surveys and experiments.[1]

4. • When census data cannot be collected,
statisticians collect data by developing specific
experiment designs and survey samples.
Representative sampling assures that inferences
and conclusions can reasonably extend from the
sample to the population as a whole.
An experimental study involves taking
measurements of the system under study,
manipulating the system, and then taking
additional measurements using the same
procedure to determine if the manipulation has
modified the values of the measurements. In
contrast, an observational study does not involve
experimental manipulation.

5. • Two main statistical methods are used in data
analysis: descriptive statistics, which summarize data from
a sample using indexessuch as the mean or standard
deviation, and inferential statistics, which draw conclusions
from data that are subject to random variation (e.g.,
observational errors, sampling variation).[3] Descriptive
statistics are most often concerned with two sets of
properties of a distribution (sample or population): central
tendency (or location) seeks to characterize the
distribution's central or typical value,
while dispersion (or variability) characterizes the extent to
which members of the distribution depart from its center
and each other. Inferences on mathematical statistics are
made under the framework of probability theory, which
deals with the analysis of random phenomena.

6. A standard statistical procedure involves the test of the
relationship between two statistical data sets, or a data set
and synthetic data drawn from idealized model. A
hypothesis is proposed for the statistical relationship
between the two data sets, and this is compared as
an alternativeto an idealized null hypothesis of no
relationship between two data sets. Rejecting or disproving
the null hypothesis is done using statistical tests that
quantify the sense in which the null can be proven false,
given the data that are used in the test. Working from a null
hypothesis, two basic forms of error are recognized: Type I
errors (null hypothesis is falsely rejected giving a "false
positive") and Type II errors (null hypothesis fails to be
rejected and an actual difference between populations is
missed giving a "false negative").[4] Multiple problems have
come to be associated with this framework: ranging from
obtaining a sufficient sample size to specifying an adequate
null hypothesis.[citation needed]

7. • Measurement processes that generate statistical
data are also subject to error. Many of these
errors are classified as random (noise) or
systematic (bias), but other types of errors (e.g.,
blunder, such as when an analyst reports
incorrect units) can also be important. The
presence of missing data or censoring may result
in biased estimates and specific techniques have
been developed to address these problems.
• Statistics can be said to have begun in ancient
civilization, going back at least to the 5th century
BC, but it was not until the 18th century that it
started to draw more heavily
from calculus and probability theory.

8. PROBABILITY
• Probability is the measure of the likelihood that
an event will occur.[1] Probability is quantified as a
number between 0 and 1, where, loosely speaking,[2] 0
indicates impossibility and 1 indicates certainty.[3][4] The
higher the probability of an event, the more likely it is
that the event will occur. A simple example is the
tossing of a fair (unbiased) coin. Since the coin is fair,
the two outcomes ("heads" and "tails") are both
equally probable; the probability of "heads" equals the
probability of "tails"; and since no other outcomes are
possible, the probability of either "heads" or "tails" is
1/2 (which could also be written as 0.5 or 50%).

9. • These concepts have been given
an axiomatic mathematical formalization
in probability theory, which is used widely in
such areas of
study as mathematics, statistics, finance, gamblin
g, science (in particular physics), artificial
intelligence/machine learning, computer
science, game theory, and philosophy to, for
example, draw inferences about the expected
frequency of events. Probability theory is also
used to describe the underlying mechanics and
regularities of complex systems.[5]
• https://en.wikipedia.org/wiki/Probability

10. STATISTICS AND PROBABILITY
• Statistics and probability are sections of
mathematics that deal with data collection
and analysis. Probability is the study of chance
and is a very fundamental subject that we
apply in everyday living, while statistics is
more concerned with how we handle data
using different analysis techniques and
collection methods. These two subjects always
go hand in hand and thus you can't study one
without studying the other.

11. Introduction to Probability
Distributions - Random Variables
• A random variable is defined as a function that associates a real
number (the probability value) to an outcome of an experiment.
• In other words, a random variable is a generalization of the
outcomes or events in a given sample space. This is possible since
the random variable by definition can change so we can use the
same variable to refer to different situations. Random variables
make working with probabilities much neater and easier.
• A random variable in probability is most commonly denoted by
capital X, and the small letter x is then used to ascribe a value to
the random variable.
• For examples, given that you flip a coin twice, the sample space for
the possible outcomes is given by the following:
S = { H H, H T, T H, T T }

12. • There are four possible outcomes as listed in the
sample space above; where H stands for heads
and T stands for tails.
• The random variable X can be given by the
following: HH
X = HT
TH
TT
• To find the probability of one of those out comes
we denote that question as: P( X = x)

13. • which means that the probability that the
random variable is equal to some real
number x.
• In the above example, we can say:
• Let X be a random variable defined as the
number of heads obtained when two coins are
tossed. Find the probability the you obtain
two heads.
• So now we've been told what X is and that x =
2, so we write the above information as:
P(X=2)

14. • Since we already have the sample space, we
know that there is only one outcomes with two
heads, so we find the probability as:
P(X = 2) = ¼
we can also simply write the above as:
P(X) = ¼
• From this example, you should be able to see that
the random variable X refers to any of the
elements in a given sample space.

15. • There are two types of random variables: discrete
variables and continuous random variables.
Discrete Random Variables
• The word discrete means separate and individual. Thus
discrete random variables are those that take on
integer values only. They never include fractions or
decimals.
• A quick example is the sample space of any number of
coin flips, the outcomes will always be integer values,
and you'll never have half heads or quarter tails. Such a
random variable is referred to as discrete. Discrete
random variables give rise to discrete probability
distributions.

16. Continuous Random Variable
• Continuous is the opposite of
discrete. Continuous random
variables are those that take on
any value including fractions and
decimals. Continuous random
variables give rise to continuous
probability distributions.

17. Sample Problem for Random Variables
Problem 1:
A couple has three kids. What will be the values which can be
attained by the random variable representing number of
daughters the couple is having?
Solution:
Let the random variable be X.
X= Number of daughters the couple is having
• The discrete values for X will be {0, 1, 2, 3} as out of three
kids there can be none, one, two or all three daughters.

18. Sample Problem 2:
From the given random variables, find out
which ones are discrete and which are
continuous.
a) Speed of train
b) Number of students getting A grade
c) Height of men in Alaska
d) Error in measurement

19. Solution:
The random variables can be classified as given
here,
• a) Speed of train will have continuous values and
hence, it is a continuous random variable.
• b) Number of students will have countable values
and hence, it is a discrete random variable.
• c) Height of men will have continuous values and
hence, it is a continuous random variable.
• d) Error in measurement will have continuous
values and hence, it is a continuous random
variable.