Probability Density Function:

Learning probability density functions:

In a normal distribution, curve quantity is y, which is represented the height f the curve at any point along x axis is called Probability density function. Here the probability density function represented using the intervals. The probability density function denoted by f(x) this function is also known as the probability distribution function and mass function, it will be referred as the probability density function(pdf).

Learning of probability density functions helps to know about the density functions of probability. The learning of the these functions are usually done in higher level of education. Learning the function of Probability density is easy when the concept is understood very clearly.

Learning Probability Density Function:

The probability of a random variable declining within a certain set is given by the integral of its density over the set. The continuous random variable is a x, and then the probability density function of x is a f(x) such that for two numbers a and b with a b

Probability distribution function is used to denote the probability density function. It may refer to the cumulative distribution function, or it may be a probability mass function rather than the density.

Formula:

A random variable which has a usual distribution with a mean m=0 and a standard deviation σ=1 is referred to as Standard Normal Distribution.

Normal Distribution = P(x) = (1/ (σ √(2π))) e^{-(x-m)2 / (2σ2)}

Standard Normal Distribution = P(x) = (1/√(2π)) e^{-(x2 / 2)}

Here m = mean σ is standard deviation, = 3.14 e= 2.718.

The normal distribution, also recognized as the Gaussian distribution, is the most widely-used common purpose distribution. It is for this reason that it is included among the lifetime distributions generally used for reliability and life data analysis.

How to Find Probability Density Function:

Solve: mean m=5

Standard deviation σ=4
Normal random variable x=6 find the probability function

Solution:

• Step 1: To calculate PDF find sqrt (2π).

sqrt (2π) = sqrt (2 x 3.14)

= sqrt (6.28) = 2.51

• Step 2: Find 1/ (σsqrt (2π)).

σsqrt (2π) = 4 x 2.51 = 10.04
1/(σsqrt(2π)) = 1/10.04 = 0.099

• Step 3: To Find e^{-(x-m) 2 / (2σ2)} and calculate -(x-m) ² and 2σ².

-(x-m) ² = - (6-5)²
= 1² = 1
2σ² = 2 x (4²)
= 2 x 16 = 32
-(x-m) ² / (2σ²) = 1/32
= 0.03125

• Step 4: Calculate e^{-(x-m) 2 / (2σ2)}

= 2.718^0.03125

= 1.0317

• Step 5: To find PDF formula is used.

= 0.099 x 1.0317

= 0.1021

Hope you like the above example of Probability Density Function.Please leave your comments, if you have any doubts.

Scale Drawing Math

Scale Drawing Math:

Scale drawing math is a type of geometry is the foundation of geometric principle of relationship. Two figures are similar if that is the same shape even though they may have different sizes. Any figure is a similar to itself, so, in this specific case, the similar figures do actually have the same size as well as the same shape.In this article we shall discuss about the scale drawing math.

Can you predict how big an image will be as you change the distance between the projector and the screen?”

Have the students talk about what they know about using an overhead projector and what approach might they use to solve the problem of scale drawing Problem.

• Place a representation on the overhead and place the projector as close to the screen as possible.
• Measure the breadth of the representation and the space between the screen and the overhead. Verification the result values.
• Move the projector backside and do again step 2. Move the overhead more than a few times and verification the results.
• Using the data, encourage to the students to calculate the ratio.
• Return the projector to an original location, replace the image and measure.
• Dialogue with a students whether they could determine the width of the image.
• See check for correctness by using the relation from the previous image.

Scale Drawings Math Lesson Example 1:

The length of an object is drawn to scale. The scale of the drawing is 1:30.The object length of sketch on the paper is 12 inches, to calculate the object in real life.

Solution:

Length of a drawing 1

----------------------------- =---

Real length 30

Do a cross product by multiply the numerator of one fraction by the denominator of the other fraction

We get:

Length of drawing × 30 = Real length × 1

length of drawing = 12, we get:

12 × 30 = Real length × 1

360 inches = Real length.

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Multiplication

Multiplication:

It is usually easy to identify a Multiplication Problem because of the symbol that is used.
Multiplication (symbol "×") is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic (the others being addition, subtraction and division).Because the result of scaling by whole numbers can be thought of as consisting of some number of copies of the original, whole-number products greater than 1 can be computed by repeated addition; for example, 3 multiplied by 4 (often said as "3 times 4") can be calculated by adding 4 copies of 3 together:

3 times 4 = 3 + 3 + 3 + 3 = 12.
3 times 4 = 4 + 4 + 4 = 12.

Multiplication is the one of the type of basic arithmetic operations. Usually addition,subtraction ,multiplication and division are the basic operations. Here we are going to discuss about multiplication operation. ‘x’ or‘.’ is the operator to identify the multiplication symbol. It is used to multiply two or more numbers.Let us now describe the steps involved in multiplication.


Brief Description about Multiplication:

In multiplication we can generally use three terms. They are

* Multiplicand
* Multiplier
* Product

For Example 2 x 3 =6

Here 2 is known as Multiplicand

And 3 is known as Multiplier

6 is known as the product

The other words of multiplication:

Symbol form:

. =Dot

X =Cross

Word Form:

* Product
* Multiply
* Times
* reciprocal of division

General Rule:

* Every number multiplied with zero is resultant to zero.
* Every number multiplied with one is resultant to the same number.

Example Problems

Example 1:

Find the multiplication value of 16 x 2

Solution:

16 times 2 is 32

Here we can use the multiplication table 2

Example 2:

Find the multiplication value of 23 x 14

Solution:

This involves the following steps.

They are

Step 1: This is the two digit multiplication.

Step 2: first multiply the 23 x 4

Step 3: we can use the multiplication table 4

Step 4:take the unit digit of 3

Step 5: that is 3 times 4 = 12

Step 6: keep 2 and consider 1 as the remainder

Step 7: then 2 times 4 is 8

Step 8: add remainder 1 to the value 8

Step 9: then it is 9

Step 10: then the resultant value of 23 x 4 is 92.

Step 11: now consider 23 x 1

Step 12: again take the unit digit value 3 x 1

Step 13: it is 3 then the remainder value is zero.

Step 14: then 2 x 1 is 2

Step 15: therefore 23 x1 = 23

Step 16: Now add both values in the following format.

Step 17: therefore the multiplication value of 23 x 14 is 322.

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Theorem:

Theorem:

Let us learn about theorems and also about the procedure how a theorem is proved.A theorem is a generalised statement, which can be proved logically. A theorem has two parts, a hypothesis, which states the given facts and a conclusion which states the property to be proved. The two statements given above are examples of theorems.

Theorems are proved using undefined terms, definitions, postulates and occasionally some axioms from algebra.

In mathematics, a theorem is a statement which has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules. Theorems have two components, called the hypotheses and the conclusions.

A theorem is a generalised statement because it is always true. For example the statement or the proposition “If two straight lines intersect, then the vertically opposite angles are equal” is true for any two straight lines intersecting at a point. Such a statement is called the general enunciation.

In the theorem stated above, “two lines intersect” is the hypothesis and “vertically opposite angles are equal” is the conclusion. It is the conclusion part that is to be proved logically. To prove a theorem is to demonstrate that the statement follows logically from other accepted statements, undefined terms, definitions or previously proved theorems.

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Axioms

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.

In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Unlike theorems, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems)

The axiom is a mathematical term which has the statements of the starting points in the geometry. It can be mathematically derived by some principles as shown in the following. This can be logically demonstrated by some mathematical proofs as follows. In geometry the axioms has the symmetric property which has the addition, subtraction, multiplication and division.Let us also learn about the various types of Axioms.

Types of Axioms:

There are two types of Axioms:

· Logical Axiom

· Non – Logical Axiom

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Ordinary Differential Equation:

Ordinary Differential Equation:

Definition:

An equation involving independent variables dependent variable and their derivatives is called a differential equation.

A differential equation which involves only one independent variable is called an ordinary differential equation.
Order of a differential Equation:

* is the order of the derivative of the highest order, occurring in the differential equation.

Degree of a differential equation:

Let us now learn about the degree of a differential equation:

* The degree of a differential equation is the degree of the highest order differential coefficient appearing in it, after all the differential coefficients are free from radical powers.

* To form a differential equation from a given equation in x, y and containing arbitrary constants. The given equation is differentiated successively as many times as the number of arbitrary constants. These equations are used to eliminate the arbitrary constants and the equation obtained is the required differential equation.

Solution of a differential equation:

* A functional relation between x and y which satisfies the given differential equation.

* Solution of a differential equation by the method of variables separable.

Step I :

Express the differential equation in the form f(x) dx = g(y) dy.
Step II :

Integrating both sides

we get the solution.

Step I:

Put ax + by + c = t

Step II:

Substituting in the differential equation

Step III:

Reducing to variables separable

Step IV:

Integrating we get the solution.

Step III:

Substituting this in the differential equation, it reduces to the form f(v) dv = g(x) dx.
Step IV:

Solution is obtained by integrating both sides and substituting

Step I:

Identify P and Q such that P and Q are functions of x only.

Step III:

Solution is obtained from

Step I:

Integrating once we get is the solution of the given Differential Equation.

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Introduction of Analytic Function

Let us learn about Analytic Function,

A complex function is said to be analytic on a region R if it is complex differentiable at every point in R. The terms holomorphic function, differentiable function, and complex differentiable function are sometimes used interchangeably with "analytic function" (Krantz 1999, p. 16). Many mathematicians prefer the term "holomorphic function" (or "holomorphic map") to "analytic function" (Krantz 1999, p. 16), while "analytic" appears to be in widespread use among physicists, engineers, and in some older texts (e.g., Morse and Feshbach 1953, pp. 356-374; Knopp 1996, pp. 83-111; Whittaker and Watson 1990, p. 83).

If a complex function is analytic on a region R, it is infinitely differentiable in R. A complex function may fail to be analytic at one or more points through the presence of singularities, or along lines or line segments through the presence of branch cuts.

A complex function that is analytic at all finite points of the complex plane is said to be entire. A single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities goes to infinity like a polynomial (i.e., these exceptional points must be poles and not essential singularities), is called a meromorphic function.
Hope the above explanation helped you.