Normal Random variable and Normal distribution
The random variable with uncountable set of values is known to be continuous random variable, and the probability density function with the help of integration as area under the curve gives the continuous distribution, Now we will focus one of the most used and frequent continuous random variable viz normal random variable which has another name as Gaussian random variable or Gaussian distribution.
Topics on Probability
Normal random variable
Normal random variable is the continuous random variable with probability density function
having mean μ and variance σ2 as the statistical parameters and geometrically the probability density function has the bell shaped curve which is symmetric about the mean μ.
We know that probability density function has the total probability as one so
by putting y= (x-μ)/σ
this double integration can be solved by converting it into polar form
which is the required value so it is verified for the integral I.
- If X is normally distributed with parameter μ and σ2 then Y=aX+b is also normally distributed with the parameters aμ+b and a2μ2
Expectation and Variance of Normal Random variable
The Expected value of the normal random variable and the variance we will get with the help of
where X is normally distributed with the parameters mean μ and standard deviation σ.
since mean of Z is zero so we have the variance as
by using integration by parts
for the variable Z the graphical interpretation is as follows
and the area under the curve for this variable Z which is known as standard normal variable, it is calculated for the reference (given in the table), as the curve is symmetric so for negative value the area will be same as that of positive values
since we have used the substitution
Here keep in mind that Z is standard normal variate where as continuous random variable X is normally distributed normal random variable with mean μ and standard deviation σ .
So to find the distribution function for the random variable we will use the conversion to the standard normal variate as
for any value of a.
Example: In the standard normal curve find the area between the points 0 and 1.2.
If we follow the table the value of 1.2 under the column 0 is 0.88493 and value of 0 is 0.5000 ,
Example: find the area for the standard normal curve within -0.46 to 2.21.
From the shaded region we can bifurcate this region from -0.46 to 0 and 0 to 2.21 because the normal curve is symmetric about y axis so the area from -0.46 to 0 is same as the are from 0 to 0.46 thus from the table
so we can write it as
Total Area =(area between z = -0.46 and z=0 ) + (area between z =0 and z=2.21)
= 0.1722 + 0.4864
Example: If X is normal random variable with mean 3 and variance 9 then find the following probabilities
Solution: since we have
so bifurcating into the intervals -1/3 to 0 and 0 to 2/3 we will get the solution from the tabular values
=0.74537 -1 + 0.62930 =0.37467
Example: An observer in paternity case states that the length (in days) of human growth
is normally distributed with parameters mean 270 and variance 100. In this case the suspect who is father of the child provided the proof that he was out of the country during a period that started 290 days before the birth of the child and ended 240 days earlier the birth. Find the probability that the mother could have had the very long or very short pregnancy indicated by the witness?
Let X denote the normally distributed random variable for gestation and consider the suspect is the father of the child. In that case the birth of the child happened within the specified time has the probability
Relation between Normal random variable and Binomial random variable
In case of Binomial distribution the mean is np and the variance is npq so if we convert such binomial random variable with such mean and standard deviation having n very large and p or q are very small going nearer to zero then standard normal variable Z with the help of these mean and variance is
here in terms of Bernouli trials X considers the number of successes in n trials. As n is increases and goes nearer to infinity this normal variate goes in the same way to become standard normal variate.
The relation of binomial and standard normal variate we can find with the help of following theorem.
DeMoivre Laplace limit theorem
If Sn denotes the number of successes that occur when n independent trials, each resulting in a success with probability p , are performed, then, for any a < b ,
In other words
Example: With the help of normal approximation to the binomial random variable find the probability of occurrence of 20 times tail when a fair coin tossed 40 times.
Solution: Suppose the random variable X represents the occurrence of tail, since the binomial random variable is discrete random variable and normal random variable is continuous random variable so to convert the discrete into the continuous, we write it as
and if we solve the given example with the help of binomial distribution we will get it as
Example: To decide the efficiency of a definite nourishment in decreasing the extent of cholesterol in the blood circulation, 100 people are placed on the nourishment. The cholesterol count were observed for the define time after providing the nourishment. If from this sample 65 percent have low cholesterol count then nourishment will be approved. What is the probability that the nutritionist approves the new nourishment if, actually, it has no consequence on the cholesterol level?
solution: Let the random variable express the cholesterol level if down by the nourishment so the probability for such random variable will be ½ for each person, if X denotes the low level number of people then the probability that result approved even there is no effect of nourishment to reduce the level of cholesterol is
In this article the concept of continuous random variable namely normal random variable and its distribution with probability density function were discussed and the statistical parameter mean, variance for the normal random variable is given. The conversion of normally distributed random variable to the new standard normal variate and area under the curve for such standard normal variate is given in tabulated form one of the relation with discrete random variable is also mentioned with example ,if you want further reading then go through:
Schaum’s Outlines of Probability and Statistics
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