Over 9,508,000 live tutoring sessions served! Sales Toll Free No: 1-866-617-602 0

Home How it works About Us Algebra Booster ! Math ▶ Statistics ▶ Probability Distribution ▶ Normal Distribution 

# Normal Distribution 0



# I need help with "Normal Distributi o Get a Tutor Probability is an essential and a very special branch of mathematics. In probability, we study about the probable chances of happening of a certain event. In the language of probability, an event is an outcome of an experiment or a process. Probability usually deals with the events which have two possible results or outcomes, such as - either pass or fail, either success or failure, either true or false, either one or zero etc. Probability of an event must lie between 0 to 1. A definite or sure event has a probability of 1; while an impossible event has a probability zero. Probability distribution is an important term that assigns the probabilities for the all possible outcomes for a finite set of events. There are different types of probability distributions. Normal distribution is most common of all. The normal distribution refers to type of the family of a continuous probability distributions defined in terms of the normal equation. Normal distribution is denoted by the equation with the random variable Y is - Y = Where, is the mean, X is a random normal variable,

referred to the standard deviation, the value of is approximately 3.14, also eis approximately 2.71828. The random variable X defined in above normal equation is known as the normal random variable. 

# exp (( ) 1 σ 2 π

√ − 1 2 x − μ σ 

# ) 2
μ σ π Normal Distribution Definition Standard Normal Distribution Normal Probability Distribution Normal Distribution Curve Properties of Normal Distribution Inverse Normal Distribution Log Normal Distribution Bivariate Normal Distribution Multivariate Normal Distribution Cumulative Normal Distribution Normal Distribution Standard Deviation Truncated Normal Distribution Half Normal Distribution Skew Normal Distribution Normal Distribution Percentile Normal Distribution Example 

### Related Concepts Confidence Interval Normal Distribution Normal Approximation Binomial Distribution Distribution Probability Beta Probability Distribution Binomial Distribution Probability Binomial Distribution Probability Examples Binomial Distribution Probability Formula Conditional Probability Distribution 

### Related Formulas Formula Normal Distribution A Probability Distribution is an Equation that Normal Distribution 

# http://math.tutorvista.com/statistics/normal-distribution.html

10/29 /201 4

### Normal Distribution Definition Back to To p



### Standard Normal Distribution Back to To p



### Normal Probability Distribution Back to To p The normal distribution is a probability distribution that associates the normal random variable X with a cumulative probability. The term bell curve is used to describe the mathematical concept called normal distribution. A bell curve graph depends on two factors, the mean and the standard deviation. The mean identifies the position of the center and the standard deviation determines the the height and width of the bell. All normal distributions look like a symmetric, bell-shaped curve.
The normal distribution, also called as Gaussian distribution, is a continuous distribution of great importance. A standard normal distribution is a normal distribution with mean 0 and standard deviation 1. The Normal distribution is often called as the "bell curve", when the random variable takes the values in the set of real numbers. Standard normal distribution is a special case of the normal distribution, also called Z distribution. The pd f of the standard normal distribution is same as normal distribution, except that it has = 0 and = 1

Normal distributions can be transformed to standard normal distributions by the formula:

z = where, X is a score from the original normal distribution, is the mean of the original normal distribution, and is the standard deviation of original normal distribution.

The normal distribution is a continuous probability distribution. This has several implications for probability. 1. The total area under the normal curve is equal to 1. 2. The probability that a normal random variable X equals any particular value is 0. 

# Free Math Tutor Online Get Answers to all Math Problems with Expert Online Tutors, Try Now

μ σ



### X

− μ

σ μ σ Binomial Distribution Formula Example Chi Square Distribution Formula 

### Related Calculators Cumulative Normal Distribution Calculator Normality Calculations Area under Normal Curve Calculator 

### Related Worksheets Normal Distribution Word Problems Normal Distribution Worksheet Standard Normal Distribution Worksheet Binomial Distribution Worksheet 

# http://math.tutorvista.com/statistics/normal-distribution.html

10/29 /201 4

### Normal Distribution Curve Back to To p



### Properties of No rmal Distribution Back to To p



### Inverse Normal Distribution Back to To p 3. The probability that X is greater than aequals the area under the normal curve bounded by aand plus infinity (as indicated by the non-shaded area in the figure below). 4. The probability that X is less than aequals the area under the normal curve bounded by aand minus infinity (as indicated by the shaded area in the figure below).

Additionally, every normal curve (regardless of its mean or standard deviation) conforms to the following rules: 1. About 68% of the area under the curve falls within 1 standard deviation of the

mean. 2. About 95% of the area under the curve falls within 2 standard deviations of the

mean. 3. About 99.7% of the area under the curve falls within 3 standard deviations of the

mean. The normal distribution graph depends on two factors - the mean and the standard deviation. The mean of the distribution determines the location of the centre of the graph, and the standard deviation determines the height and width of the graph. When the standard deviation is large, the curve is short and wide. When the standard deviation is small, the curve is tall and narrow. Normal Distribution Bell Curve All normal distributions looks like a symmetric, bell-shaped curve, as shown below. The curve on the left is shorter and wider than the curve on the right, because the curve on the left has a bigger standard deviation.

Some properties of the normal distribution: • The normal curve is a symmetric bell-shaped curve. • 50% of the scores lie above and 50% below the midpoint of the distribution. • The total area under the curve is equal to 1. • Curve is asymptotic to the x axis • Mean, median, and mode are equal and located at the midpoint of the x axis. The inverse normal probability distribution function will find the precise value at a given percent based upon the mean and standard deviation. the inverse Gaussian distribution is a two-parameter family of continuous probability distributions with support on (0, ). Its probability density function is given by

= for and . 

### ∞ f ( x , μ , λ ) 

# [ exp λ 2 πx 3 

# ] 1/2

− λ ( x − μ ) ) 2 

### 2 x

μ 2


### x > 0, μ > 0 λ > 0 

# http://math.tutorvista.com/statistics/normal-distribution.html

10/29 /201 4

### Log Normal Distribution Back to To p



### Bivariate Normal Distribution Back to To p



### Multivariate Normal Distribution Back to To p



### Cumulative Normal Distribution Back to To p



### Normal Distribution Standard Deviation Back to To p where, is the mean and is the shape parameter. A continuous distribution in which the logarithm of a variable has a normal distribution. Log-normal distributions can model a random variable X where log(X) is normally distributed. The probability density functions for the log normal distribution is

=

The bivariate normal distribution is the statistical distribution with probability density function. The bivariate normal distribution has two variables, and , possibly dependent.
= where is correlation of and .
The multivariate normal distribution model extends the univariate normal distribution model to fit vector observations. A p-dimensional vector of random variables, , is said to have a multivariate normal distribution if its density function is of the form

= The cumulative distribution function describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. A cumulative distribution function gives the cumulative area with a pdf. The shape of the various distribution cdfs are distinctive, and the cdf is merely another way of illustrating the distribution. The cumulative distribution function is probability of a value less than x.

For the normal distribution, this gives in terms of the complimentary error function by
= 

### μ λ

P ( x ) 

# exp 1 s x 2 π √ −( lnx

− M) 2 

### 2 s 2



### x 1 

### x 2 

### P (, )

x 1 

### x 2 

# exp [ ] 1 2 πσ 1 

### σ 2 

### 1− ρ 2 

### √ − z 2(1− ) ρ 2 

### ρ x 1 

### x 2



### X = , ,..., X1 

### X 2 

### X p 

### −∞ < < ∞ X

i 

### i = 1, . . . , p f ( X ) f ( x ) 

# ( | ∑ exp ( ( X

− m ) ( X

− m )) 1 2 π 

# ) ρ 2 

# | 1/2

− 1 2 

# ∑ −1

cdf = P (< x ) = P ( x ) dx ∫ x −∞ 

# erfc ( ( )) 1 2 − 1 2 √ x − μ σ 

# http://math.tutorvista.com/statistics/normal-distribution.html

10/29 /201 4

### Truncated Normal Distribution Back to To p



### Half Normal Distribution Back to To p



### Skew Normal Distribution Back to To p The Normal distribution is often called as the "bell curve", when the random variable takes the values in the set of real numbers. Normal Distribution Formula The normal distribution is defined by the equation with the random variable is
= where, is a normal random variable, is the mean, is the standard deviation. The distribution with and is called the standard normal distribution or the unit normal distribution.

The truncated normal distribution assumed for x generalizes the one parameter half normal distribution, by allowing the normal distribution, which is truncated below at zero, to have a nonzero mode. The truncated normal distribution contains an additional parameter to be estimated and so provides a more flexible representation of the pattern of the efficiency in the data. The truncated normal density function for is given by

=
The half-normal distribution is a normal distribution with mean 0 and parameter limited to the domain, lies between to . It has probability and distribution functions given by
= , . The half-normal distribution is a special case of the truncated normal distribution, with the estimated being mode 0. The probability density function for the half-normal distribution with variance parameter , in the domain as defined as

= The skew normal distribution is a continuous probability distribution that generalized the normal distribution to allow for non-zero skewness. The skew-normal distribution has density as follows: 

### f ( x , μ , ) σ 2 

### exp (( ) 1 σ 2 π √ − 1 2 x − μ σ 

### ) 2 

### x μσ μ =0 =1 σ 2



### μ f ( x ) x ≥ 0 f ( x ) 

# exp (− ) 1 σφ ( ) 2 π √ μ σ 

### ( x − μ ) 2 

### 2 σ 2



### x 0 ∞
f ( x , σ ) 

# exp ( ) 2 √ σ π

√ − y 2 

### 2 σ 2 

### y > 0

θ 0 ≤ x ≤ ∞ f ( x ) 

# exp ( ) 2 θ π − θ 2 

### x 2 

### π 

# http://math.tutorvista.com/statistics/normal-distribution.html

10/29 /201 4

### Normal Distribution Percentile Back to To p



### Normal Distribution Example Back to To p = for where is the standard normal cdf, and is constant. • For = 0, the skew-normal distribution is the normal

distribution, and if | | is relatively small, the distribution is close to the normal. • For large | |, the distribution is more skewed.

The percentile of a normal distribution with mean and standard deviation is easily related to the percentile of the standard normal distribution. An important aspect of the normal distribution is that we can easily find the percentiles of the distribution. Percentile for normal( ) = + (Percentile for standard normal) If z is the desired percentile for the standard normal distribution, then the desired percentile for the normal distribution is z standard deviation from . Listed below are some of the normal distribution examples. Solved Examples Question 1: An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assuming that bulb life is normally distributed, what is the probability that an Acme light bulb will last at most 365 days? Solution: Given a mean score of 300 days and a standard deviation of 50 days, we want to find the cumulative probability that bulb life is less than or equal to 365 days. Thus, we know the following: 1) The value of the normal random variable is 365 days. 2) The mean is equal to 300 days. 3)The standard deviation is equal to 50 days. We enter these values into the No rmal Distribution Calculator and compute the cumulative probability. The answer is: P( X < 365) = 0.90. Hence, there is a 90% chance that a light bulb will burn out within 365 days.

Question 2: Suppose scores on an IQ test are normally distributed. If the test has a mean of 100 and a standard deviation of 10, what is the probability that a person who takes the test will score between 90 and 110? Solution: Here, we want to know the probability that the test score falls between 90 and 110. The "trick" to solving this problem is to realize 

### f ( x ) 

# exp ( ) φ ( λx ) 2 2 π √ − x 2 

### 2 −∞ ≤ x ≤ ∞ φ λ λ λ λ

μ σ

μ , σμ σ μ 

# http://math.tutorvista.com/statistics/normal-distribution.html

10/29 /201 4◀◀ Binomial Distribution Bernoulli Distribution ▶ ▶ the following: P( 90 < X < 110 ) = P( X < 110) - P( X < 90) We use the Normal Distribution Calculator to compute both probabilities on the right side of the above equation. 1) To compute P( X < 110), we enter the following inputs into the calculator: The value of the normal random variable is 110, the mean is 100, and the standard deviation is 10. We find that P( X < 110) is 0.84. 2) To compute P( X < 90), we enter the following inputs into the calculator: The value of the normal random variable is 90, the mean is 100, and the standard deviation is 10. We find that P( X < 90) is 0.16. We use these findings to compute our final answer as follows: P( 90 < X < 110 ) = P( X < 110) - P( X < 90) P( 90 < X < 110 ) = 0.84 - 0.16 P( 90 < X < 110 ) = 0.68 Thus, about 68% of the test scores will fall between 90 and 110.



### More topics in Normal Distribution Z Score Table 

### Related Topics Math Help Online Online Math Tutor *AP and SAT are registered trademarks of the College Board. About Us | Careers | Contact Us | Blog | Homework Help | Teaching Jobs | Search Lessons | Answers | Cal culators | Worksheets | Formulas Copyright © 2014 - NCS Pearson, All rights reserved. 

# http://math.tutorvista.com/statistics/normal-distribution.html

10/29 /201 4