A variety of mathematical aids to probability analysis and calculations.
Series
Geometric series From the expression
, we obtain
For
, these sums converge to the geometric series
Differentiation yields the following two useful series:
For the finite sum, differentiation and algebraic manipulation yields
Exponential series .
Simple algebraic manipulation yields the following equalities usefulfor the Poisson distribution:
Sums of powers of integers
Some useful integrals
The gamma function Integration by parts shows
By induction
For a positive integer
By a change of variable in the gamma integral, we obtain
A well known indefinite integral gives
For any positive integer
m ,
The following integrals are important for the Beta distribution.
For nonnegative integers
Some basic counting problems
We consider three basic counting problems, which are used repeatedly as components of
more complex problems. The first two,
arrangements and
occupancy are
equivalent. The third is a basic
matching problem.
Arrangements of
r objects selected from among
n distinguishable
objects.
The order is significant.
The order is irrelevant.
For each of these, we consider two additional alternative conditions.
No element may be selected more than once.
Repitition is allowed.
Occupancy of
n distinct cells by
r objects. These objects are
Distinguishable.
Indistinguishable.
The occupancy may be
Exclusive.
Nonexclusive (i.e., more than one object per cell)
The results in the four cases may be summarized as follows:
We reduce the problem to determining
, as follows:
Select
k places for matches in
ways.
Order the
remaining elements so that no matches in the other
places.
Some algebraic trickery shows that
is the integer nearest
. These are
easily calculated by the MATLAB command
M = round(gamma(n+1)/exp(1)) For example
>>M = round(gamma([3:10]+1)/exp(1));>>disp([3:6;M(1:4);7:10;M(5:8)]')3 2 7 1854
4 9 8 148335 44 9 133496
6 265 10 1334961
Extended binomial coefficients and the binomial series
The ordinary
binomial coefficient is
for integers
For any real
x , any integer
k , we extend the definition by
and
Then
Pascal's relation holds:
The power series expansion about
shows
For
, a positive integer, the series becomes a polynomial of degree
n .
Cauchy's equation
Let
f be a real-valued function defined on
, such that
, and
There is an open interval
I on which
f is bounded
above (or is bounded below).
Then
Let
f be a real-valued function defined on
such that
, and
There is an interval on which
f is bounded above.
Then, either
, or there is a constant
a such that
[For a proof, see Billingsley,
Probability and Measure , second edition, appendix A20]
Countable and uncountable sets
A set (or class) is
countable iff either it is finite or its members can be
put into a one-to-one correspondence with the natural numbers.
Examples
The set of odd integers is countable.
The finite set
is countable.
The set of all rational numbers is countable. (This is established by
an argument known as diagonalization).
The set of pairs of elements from two countable sets is countable.
The union of a countable class of countable sets is countable.
A set is
uncountable iff it is neither finite nor can be put into a one-to-one
correspondence with the natural numbers.
Examples
The class of positive real numbers is uncountable. A well known operation shows that
the assumption of countability leads to a contradiction.
The set of real numbers in any finite interval is uncountable, since these can be
put into a one-to-one correspondence of the class of all positive reals.
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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