Normal Probability Distribution

The Normal Distribution can be viewed as the limiting distribution of a binomial random variable. That is, in a binomial experiment, if we use a fixed probability of success p, we can analyze what happens as the number of trials n increases. To visualize what happens, we can construct histograms for a fixed p and increasingly large n.

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The Below figure is an example of Normal Distribution:

The normal distribution is often used to describe, at least approximately, any variable that tends to cluster around the mean. For example, the heights of adult males in the United States are roughly normally distributed, with a mean of about 70 inches (1.8 m). Most men have a height close to the mean, though a small number of outliers have a height significantly above or below the mean.


-->Importance of Normal Distribution:

• Normal Distribution will appear in the different statistical applications.
• Because Normal Distribution is the central limit theorem and it is used to define the addition of random variables and their approximately distributed in normally and the number of observation is large.
• Suppose if the distribution is not exactly normal, it is convenient to assume as a normal distribution. It is a good approximation.

Baye's Theorem

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Bayes theorem relates the marginal and conditional probabilities of events A and B, where B has a non-vanishing probability:
P (A/B) = P (B/A) P (A)
P (B)

Each term in the above Baye’s theorem has a conventional name:

• P (A) is the marginal probability or prior probability of A. It is "prior" in the sense of that it does not take into account any information about B.
• P (A|B) is the conditional probability of A, given B. It is also called the posterior probability (P) because it is derived from or depending upon the specified value of B.
• P (B|A) is the conditional probability of B given A. It is also called the likelihood.
• P (B) is the marginal or prior probability of B, and acts as a normalizing constant.

Let us understand this with the below example on Bayes Theorem:
Suppose there is a school with 60% boys and 40% girls as students. The female students wear trousers or skirts in equal numbers; the boys all wear trousers. An observer sees a (random) student from a distance; all the observer can see is that this student is wearing trousers. What is the probability this student is a girl? The correct answer can be computed using Bayes' theorem.
The event A is that the student observed is a girl, and the event B is that the student observed is wearing trousers. To compute P(A|B), we first need to know:

• P(A), or the probability that the student is a girl regardless of any other information. Since the observers sees a random student, meaning that all students have the same probability of being observed, and the fraction of girls among the students is 40%, this probability equals 0.4.
• P(B|A), or the probability of the student wearing trousers given that the student is a girl. As they are as likely to wear skirts as trousers, this is 0.5.
• P(B), or the probability of a (randomly selected) student wearing trousers regardless of any other information. Since half of the girls and all of the boys are wearing trousers, this is 0.5×0.4 + 1×0.6 = 0.8.

Given all this information, the probability of the observer having spotted a girl given that the observed student is wearing trousers can be computed by substituting these values in the formula:

Variables and their types.

Variables whose values are determined by chance are called random variables.

There are two types of variables:

1) Qualitative variables.
2) Quantitative variables.


Qualitative variables are non numeric in nature and do have any particular ordering.

here are some examples of Qualitative variables; Hair color (Black, Brown)


Quantitative variables can assume numeric values and can be classified into two groups:

• Discrete variables.
• Continuous variables.

here are some examples of Quantitative variables; height, weight etc.....

Basics of Statistics.

Statistics is about collection of information and its presentation and about drawing inferences from these. We come across facts and figures in the newspapers, Television and the radio.

Statistics can be classified as follows:

1) Descriptive Statistics.

2) Inferential Statistics.

Descriptive Statistics includes:

• Collectiing.
• Organizing
• Summarizing.
• Presenting Data.

Inferential Statistics includes:

• Making inferences.
• Hypothesis testing.
• Determining relationships.
• Making predictions.

To obtain information, data are collected from variables used to describe an event. Data is nothing but the values or measurements that variables describing an event can assume.

Ellipse Equation

Deffination: "An Ellipse is a plane curve that results from the intersection of a cone bye a plane in a way that produces a closed curve".

Below is the example of Ellipse Equation:

In analytical geomentry the ellipe can be described in locus of point bears a constant ratio. The distance form the point is called application of ellipse. Tracing of the Ellipse: `(x^2/y^2) ` + ` (y^2/b^2) ` = 1, a > b Solving.

solving equation by using FOIL method

The FOIL Method is a process used in algebra to multiply two binomials. The lesson on the Distributive Property, explained how to multiply a monomial or a single term such as 7 by a binomial such as (4 + 9x).

But, what if there was a binomial instead of a single term outside of the parentheses? That is, what if a binomial was being multiplied by another binomial? An example of this is given in the problem.
Let's see an algebra question on this .

Question:-
Solve the equations by using foil method.

(y+1)(2y-3) = 25

Answer:-

Let's see how to solve equations using FOIL method.



(y+1)(2y-3) =25

2y²-3y+2y-3=25

2y²-y-3 = 25

2y²-y-3-25=0

2y²-y-28=0

2y²-8y+7y-28=0

2y(y-4)+7(y-4)=0

(y-4)(2y+7)=0

y-4=0 or 2y+7=0

y=4 or y= -7/2

y = 4 or -7/2 is the Answer

Completing the Square Method

Sometimes the roots of a quadratic equation cannot be obtained by simple factorization. So, a more general method is used. This method, which is based on the fact that any quadratic equation may be written in the form of (x+p)² = q, where p and q are real numbers, is known as completing the square method.

Question:-

Solve the equation by completing the square method.

2x²-7x+9=(x-3)(x+1)+3x

Answer:-

2x²-7x+9 = x²+x-3x-3+3x
-x²-x+3 -x²-x+3
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x²-8x+12 = 0

now we solve this quadratic equation by completing the square method

x²-8x=-12

x²-8x+(8/2)²=(8/2)²-12

x²-8x+16 = 16-12

(x-4)² = 4

taking the square root on both sides, square root symbol looks like √ .

√(x-4)²= √4

x-4 = ±2

We also can use square root calculator to get these values.

x-4 = +2 or x-4 = -2

x=6 or x=2 is the answer

This equation have just one variable.similarly we can also work on linear equations in two variables