Normal Distribution, Business Mathematics and Statistics Video Lecture | Business Mathematics and Statistics - B Com

welcome normal distribution basics
primary level problems in normal
distribution for the students who are
beginners in studying probability
probability distribution and normal
distribution let us take the case in
which mean is given to be 21.5 this is
the sign of mean normally used in normal
distribution and that is known as mu it
is called
you so mean or mu is 21.5 the standard
deviation of the data is 2.5 this is
small Sigma there are two kinds of signs
of Sigma this is B your capital Sigma
that is sign of summation or total and
this is small Sigma which is normally
used a sign of population or normal
distribution find probability of 18 less
than or equal to X less than or equal to
25 X is the sign which is generally used
as a sign of variable or statistical
data in statistics mathematics etc here
it is sign of the same numerical data
what is the probability there a
particular moment extinct value between
18 and 25 both inclusive here X can take
value either 18 or 25 or in between now
to solve any problem of normal
distribution we have to consider
standards and scope that is standard
normal variate z said is any value of
the variable minus mean divided by
standard deviation popularly it is
spoken as X minus mu of 1 standard
one small Sigma or x- mean upon standard
deviation here X has two possible limits
so let us first take X equal to 18 so
first Z will be 18 minus 21 point 5
divided by 2 point 5 that means minus 3
point 5 divided by 2 point 5 so first
value of Z is negative 1 point 4 now
another or higher limit of X is 25 for
that Z will be 25 minus 20 1 point 5
upon 2 point 5 that is positive 3 point
5 a 12.5 so here Z takes value positive
1 point 4 now what a sketch of
probability curve or normal it is always
from negative infinity to positive
infinity the mean or the variable takes
value exactly equivalent to mean in the
middle of the curve here variable is
equivalent to me in 21.5 and therefore Z
is always 0 now we know that the
left-hand side area where X takes value
less than me is point 5 from x equals to
me to x equals to negative infinity
similarly in the right hand side area
the variable takes value greater than
mean from x equivalent to mean to
positive infinity
the area is point 5 this area itself is
probability now let us take the data of
this problem first value of Z is
- 1.4 because here X takes value less
than me so it falls in the
left-hand-side region here X takes value
18 and therefore Z comes to - 1.4 now if
X falls in this region its value will be
greater than 18 but less than me or less
than 21.5 now another way you there is
positive 1.4 that is positive because it
states value greater than me so it falls
in the right hand side region this is a
sketch so no standard distance or
anything is to be maintained but it
should be somewhat symmetric X takes
value 25 therefore Z takes value
positive 1.4 if X falls in this region X
hell value less than 25 but more than
vino 21.5 so ultimately we can interpret
that if X falls in this region this one
then only it takes value between 18 and
25 or actually if X takes value 18 to 25
both inclusive then only it falls in
this region
now so far as the normal standard normal
variate said is concerned the area
between said zero and any value of z
whether it is positive or negative area
remains
I can never be negative or positive area
is actually always positive so the area
between Z zero and particular value of Z
either left hand side the right hand
side is the probability of taking the
value in that which falls in that
particular region so if we use the
readily available papers of area under
the normal curve the area between Z 0
and Z 1.4 comes to 0.4 1 9 2 4 now here
also it is the area between sin 0 and Z
1.4 so same area of this side is also
covered point 4 1 9 2 for the
interpretation of this area is area
between Z 0 and Z minus 1 point 4 is
point 4 1 9 2 4 that means the
probability of X taking value between 18
and mean is point 4 1 9 to 4 similarly
on right hand side right here between 0
and said 1.4 is point 4 1 9 2 4 that
means the probability that X can take
value between me or 21 point 5 and 25 is
point 4 1 9 to 4 we are interested in
both the areas so logically the
probability is nothing but summation of
these two values so the probability of
18 less than or equal to X less than or
equal to 25 equals to probability minus
1 point 4 less than or equal to Z less
than or equal to positive 1 point 4 it
is not necessary every time to write
this just to under
the concept that means point 4 1 9 2 4
plus 0.4 1 9 2 4 you can also write
their area between 0 and said 1.4 plus
area between 0 and z+ 1 point 4 so it
comes to 0.8 3 8 5 4 other eight three
eight four eight so this is the
probability that X variable X can take
value between 18 and 25 if it follows
normal distribution we can also express
the probability in terms of percentage
if we just multiply the probability by
100 that's it thank you very much