The mean value and the variance give important information about the distribution for a real random variable X. We consider the expectation of an appropriate function of a pair (X, Y) which gives useful information about their joint distribution. This is the covariance function.
Covariance and the correlation coefficient
The mean value
and the variance
give
important information about the distribution for real random variable
X . Can
the expectation of an appropriate function of
give useful information about
the joint distribution? A clue to one possibility is given in the expression
The expression
vanishes if the pair is independent (and in some
other cases). We note also that for
and
To see this, expand the expression
and use linearity to get
which reduces directly to the desired expression.
Now for given
ω ,
is the variation of
X from its mean
and
is the variation of
Y from its mean. For this reason,
the following terminology is used.
Definition . The quantity
is called
the
covariance of
X and
Y .
If we let
and
be the centered random
variables, then
Note that the variance of
X is the covariance of
X with itself.
If we standardize, with
and
,
we have
Definition . The
correlation coefficient
is the quantity
Thus
. We examine these concepts for information on the joint distribution.
By Schwarz' inequality (E15), we have
Now equality holds iff
We conclude
, with
iff
Relationship between
ρ and the joint distribution
We consider first the distribution for the standardized pair
Since
we obtain the results for the distribution for
by the mapping
Joint distribution for the standardized variables
,
iff
iff all probability mass is on the line
.
iff
iff all probability mass is on the line
.
If
, then at least some of the mass must fail to be on these lines.
The
lines for the
distribution are:
Consider
. Then
. Reference to
[link] shows this is the average of the
square of the distancesof the points
from the line
(i.e., the variance about
the line
). Similarly for
,
is the variance about
. Now
Thus
is the variance about
(the
line)
is the variance about
(the
line)
Now since
the condition
is the condition for equality of the two variances.
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities.
before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these
Jharna
continue..
sometimes due to these physical problems..the diagnosis may be misdiagnosed.
treatment for autism.
well it depends on the severity.
since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child
Jharna
child interact with other kids under doc supervision.
play therapy.
speech therapy.
Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game.
string and beads game.
the more you interact with the child the more effective
Jharna
results you'll get..
please consult a therapist to know what suits best on your child.
and last as a parent. I know sometimes it's overwhelming to guide a special kid.
but trust the process and be strong and patient as a parent.
Jharna
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