This document discusses continuous probability distributions, including the normal and exponential distributions. It defines a continuous random variable and probability density function. The key properties of the normal distribution are described, such as its bell-shaped, symmetric curve defined by the mean and standard deviation. Examples demonstrate how to generate random normal variables and calculate probabilities using R commands. The exponential distribution is also introduced, which has a positively skewed density curve where the mean and standard deviation are equal.

2. CONTINUOUS PROBABILITY DISTRIBUTIONS
A continuous random variable assumes any value in an interval . For large
number of observations in a data set, the histogram can be approximated by a
smooth curve.This histogram has following two properties:
(I ) The total area under the histogram is one and
(ii) For two points , say a & b ,such that each is a boundary point of some class,
the relative frequency of measurements in the interval [a ,b ] is the area under
the histogram enclosed by this interval.
If we increase the number of observations & decrease width of the class interval ,
then the jumps between rectangles will dampen out and the histogram will
approximate the shape of a smooth curve that is superimposed.The
mathematical function that produces this smooth curve is known as probability
density function of the random variable X.

3. 1.
2.The total area under the density curve is one.
When determining the probability of an interval, we need not be
concerned with whether either of the end points is included in the
interval. So,
= area under the curve between the ordinates
at a and b.
Probability density function
We define the probability density function f(x) of a continuous random
variable X as a function satisfying the following properties:

4. SOME SPECIAL CONTINUOUS DISTRIBUTIONS
In this section we discuss two commonly used continuous
distributions , namely normal distribution and exponential
distribution.
NORMAL DISTRIBUTION
A random variable X is said to have a normal distribution with
parameters (called mean)and (variance) if it’s pdf is given by
,
=
x follows normal distribution with parameters and is denoted
by .
If mean is zero and sd is 1 then x is said To have standard normal
distribution.
=

5. Properties of the normal distribution :
1.It is a continuous distribution described by a bell shaped symmetric curve.
2. Mean of the distribution is the parameter and standard deviation of
the distribution
3.It is a unimodal distribution and by symmetry of the distribution ,
mean=mode=median.
4.The distribution is completely determined by it’s parameters and .
5.The mean determines where the center of the distribution is located.
6.The standard deviation determines the flatness of the distribution.position
of the center does not change if only standard deviation is changed.

6. R- commands for normal distribution:
1.For generating a random sample of size nn from normal distribution with
specified values of mean and sd,we use rnorm function by giving
command :rnorm(x,mean,sd).
2.To find ordinate at x of normal distribution for specified values of the
parameters,we use dnorm function by giving command
dnorm(x,mean,sd).
3.To evaluate cdf at x of normal distribution with specified values of the
parameters ,we use pnorm function by giving command pnorm
(x,mean,sd).This gives area to the left of x for the specified normal
distribution.
4.To obtain p-th quantile (x) of normal distribution for specified values of
parameters,we use qnorm function by giving command qnorm(p , mean ,
sd). ie , x is such that .

7. eg 1) Draw a random variable of size 20 from N(5,2) distribution. Find
mean ,median and sd of the sample.
Soln :> x<-rnorm(20,5,2) >
> x
[1] 2.821134 7.310583 5.348927 5.753956 5.266179 5.676851 6.763142
3.419732
[9] 4.504724 5.668883 6.130423 1.727179 3.848170 7.453556 4.097615
1.857928
[17] 4.113022 2.774420 7.612387 3.062009
> smean<-mean(x)
> smean
[1] 4.890009
> smedian<-median(x)
> smedian
[1] 3.839153
> ssd<-sd(x)
> ssd
[1] 1.709635
Mean and median are close ,indicating symmetry of the distribution.The value of the
sample mean is close to the value of population mean . The value of sample sd is
close to the value of population sd.

8. eg2)Suppose x follows standard normal distribution . Determine the following
probabilities: 1 ) 2) 3)
Soln : We use the command pnorm() for evaluation of probabilities.
1) = pnorm(2)
[1] 0.9772499
2)
= pnorm(2.5)-pnorm(.84)
[1] 0.1942445
3)
= 1-pnorm(2)
[1] 0.02275013
eg3)suppose z is a standard normal variable . In each case find c such that
, ,

9. R-commands to find c are given below.
c<-qnorm(.1151)
C is 0 .1151-th quantile of the standard normal distribution
c
[1] -1.199844
c<-qnorm(.1525+pnorm(1))
> c
[1] 2.503116
 c<-qnorm((1+.8164)*1/2)

> c
[1] 1.329752
.

10. Eg4)A large scale survey conducted in a city revealed that 30% adult males
were found to be smokers . what is the probability that , in a random
sample of 1000 adults from the same city , there will be
1.More than 315 smokers?
2.Less than 280 smokers?
Soln : If x denotes the no.of smokers in the sample of 1000,then
Since n is large ,we can approximate binomial distribution by normal
distribution.
So,
To find and ,we use R-commands.
> p1<-1-pnorm(315,300,14.5)
> p1
[1] 0.1504553
> p2<-pnorm(280,300,14.5)
> p2
[1] 0.08389954

11. The normal distribution is by far the most used distribution in statistics .
There are at least 3 good reasons for it’s wide applicability.
1.Any measurement can be assumed to represent the true value plus error.
The error component is a result of pure chance . such errors in many
instances follow the normal distribution .
2.Normal distribution provides sound mathematical basis for statistical
inference.
3.The normal distribution is a good approximation to many distributions when
the sample size is large.
Importance of normal distribution:

12. EXPONENTIAL DISTRIBUTION
Dfn: A random variable X is said to have an exponential distribution with
parameter if its pdf is given by,
For this distribution ,mean and sd are equal. Exponential distribution plays an
important role in study of life -time data ,which has applications in
reliability and survival analysis. Its density curve is positively skewed.
And density curve for normal is symmetric bell-shaped curve.
Eg1) An engineer, observing a nuclear reaction ,measures time
intervals between emissions of beta particles . Following are inter
arrival times:
.894,.235,.071,.459,.1,.991,.424,.159,.431,.919,.061,.216,.082,.092,.9
,.186,.579,1.653,.83,.093,.311,.429,