Binomial Distribution :


Binomial distribution

gives us the probabilities of interests are those of receiving a certain number of success r in n independent trails each having two possible outcomes and sample probability p , of success so for example using a binomial distribution we can determine the probability of giving h heads into coin toss coin

How does the binomial distribution do this?

Basically a two part process is involved, first we have to determine the

probability of one possible way the event can occur and then determine the number of different way the event can occur that is P(Event) = (number of ways event can occur * p(one occurrence )

The Importance of Normal distribution :

1- Normal distribution is very useful because:
Many things actually are normally distributed, or very close to it. Forimages (2) example, height and intelligence are approximately normally distributed; measurement errors also often have a normal distribution
• The normal distribution is easy to work with mathematically. In many practical cases, the methods developed using normal theory work quite well even when the distribution is not normal.
• There is a very strong connection between the size of a sample N and the extent to which a sampling distribution approaches the normal form. Many sampling distributions based on large N can be approximated by the normal distribution even though the population distribution itself is definitely not normal.

Most important Characteristics of the Normal distribution :

Actually we can say that Normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a standard of reference for many probability problems So Normal distribution characteristics  is :     images (1)

• Symmetric & bell shaped
• Continuous for all values of X between -∞ and ∞ so that each conceivable interval of real numbers has a probability other than zero.
• -∞ ≤ X ≤ ∞
• Two parameters, μ and σ. Note that the normal distribution is actually a family of distributions, since μ and σ determine the shape of the distribution.
• The rule for a normal density function is e21 = )

f(x) = EXP[-(x-mu)**2/(2*sigma**2)]/(sigma*SQRT(2*PI))

where mu is the  location parameters and sigma is the scale parameters. The case where mu = 0 and sigma = 1 is called the standard normal distribution. The equation for the standard normal distribution is

f(x) = EXP[-x**2/2]/SQRT(2*PI)

• The notation N(μ, σ2) means normally distributed with mean μ and variance σ2. If we say X ∼ N(μ, σ2) we mean that X is distributed N(μ, σ2).
• About 2/3 of all cases fall within one standard deviation of the mean, that is
P(μ - σ ≤ X ≤ μ + σ) = .6826.
• About 95% of cases lie within 2 standard deviations of the mean, that is
P(μ - 2σ ≤ X ≤ μ + 2σ) = .9544

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What is a Normal Distribution?


We can say ”Normal distribution is a continuous distribution that is “bell-shaped”. Data are often assumed to be normal. Normal distributions can estimate probabilities over a continuous interval of data values.


Which Data Values Are Most Likely to be Observed in a Normal Distribution?

In a normal distribution, data are most likely to be at the mean. Data are less likely to farther away from the mean. Are the people around more likely to be short, tall, or average in height?

What is a Standard Normal Distribution?

Also we can say A standard normal distribution is a normal distribution with a mean=0 and standard deviation = 1.

standard normal

“All normal distributions can be converted into a standard normal distribution”

Why Convert to a Standard Normal Distribution?

The values for points in a standard normal distribution are z-scores. We can use a standard normal table to find the probability of getting at or below a z-score. (a percentile).

How do You Convert a Normal Distribution to a Standard Normal Distribution?

1. Subtract the mean from each observation in your normal distribution, the new mean=0.

2. Divide each observation by the standard deviation, the new standard deviation=1.