Objectives:
Conversion of a non-numeric continuous sample space to set of real numbers with the help of continuous r.v. To obtain summary statistics of a continuous r.v. To study the nature of r.v. such as central value, spread, symmetry. To develop tools such as M.G.F., C.G.F. for further study. To obtain probability of events related to continuous r.v. To obtain the probability distribution of a r.v., which is a function of r.v.
Introduction
We have seen that, there are three types of sample spaces (i) finite, (ii) countably infinite and (iii) continuous. In this chapter we discuss the random variables defined on continuous sample space.
A sample space which is finite or countably infinite is called as denumerable or countable. If the sample space i not countable then it is called continuous. In other words for a continuous sample space 2 we can not have one-to-one correspondence between 2 and set of natural numbers (1, 2,.....}. Illustrations of uncountable sample space:
(1) Suppose, weight of an oil bag having the capacity of 1 kg filled by an automatic filling machine is noted.
The sample space will be an interval in the neighbourhood of 1 kg such as (0.980, 1.005).
(2) Suppose in an experiment, life of an electronic component in hours is recorded.
The sample space in this case may be an interval as a part of R+ such as 2 = (0, 5000)
Note: A continuous sample space is a subset of real line.
Continuous Random Variable
In general, we define a random variable X () as a real valued function on domain 2. If the range set of X (w) is continuous the r.v. is continuous. The range set will be a subset of real line.
Illustrations of continuous r.v.:
(1) Weight of a person in kg.
(2) Consumption of electricity of a town in a specific month.
(3) Daily rainfall in cm. at a particular place.
(4) Instrumental error (measured in suitable units) in the measurement.
(5) Life in hours of an electrical component.
Note: The distinction between continuous random variable and discrete random variable is as follows:
(1) A continuous r.v. takes all possible values in a range set. The set is in the form of interval. On the other hand discrete r.v. takes only specific or isolated values.
(2) Since, a continuous r.v. takes uncountably infinite values no probability mass can be attached to a particular value of r.v. X. Therefore, P (X = x) = 0 for all x. However in case of a discrete r.v., probability mass is attached to individual values taken by
r.v.
In case of continuous r.v. probability is attached to an interval which is a subset of R.
Continuous Probability Distribution
In case of discrete r.v. using p.m.f. we get probability distribution of r.v., however in case of continuous r.v. probability mass is not attached to any particular value. It is attached to an interval. The probability attached to an interval depends upon its location.
For example, P (a < x < b) varies for the different values of a and b. In other words, it will not be uniform. In order to obtain the probability mass associated with any interval, we need to take into account the concept of probability density.
A function f (x) which is to be treated as a probability density function, should be a non-negative and continuous function of x. The probability that a variable X takes values in a small interval
(x-xx + x)
will be the product of length of interval and the value of density function f (x) at the centre of interval.
P()
Note: Here, we assume that the probability density is constant over the interval
(x-x, x + x).
This assumption will not be valid for
large small interval. To overcome this difficulty we integrate f (x) w.r.t. x over the given interval. (In case of discrete r.v. we take summation). Thus, b
P (a < x < b) = f(x) dx.
a
(2) The above probability is a definite integral, hence geometrically it is the area under curve y = f (x) bounded by X axis and the ordinates at a and b.
Precise definition of probability density function is as follows. Definition: A real valued function f (x) is called as a probability density function (p.d.f.) of a continuous random variable X if,
f(x) = 0; ∞ < x < ∞ (ii) f(x) dx = 1.
Note: (1) Since, probability associated with any individual value of a continuous random variable is zero, P (a < x < b) = P(a ≤ X <b) =
= P(a < x ≤ b) = P(a ≤ X ≤ b) = f (x) dx = Area under the curve f
(x) bounded between X axis and the ordinates at a and b.
It is shown in the following figure by shaded region.
(ii)0 f(x) dx, can be interpreted as P (∞ <X < ∞). It also represents the total area under the density curve bounded by X-axis.
Thus, the total area under the curve = P(- ∞ <X < ∞) = f(x) dx = 1.
Fig. 1.1
(ii) If AR, then P (X = A) = f(x) dx.
(iv) f(x) need not be less than 1. Example: Let X be r.v. with following p.d.f.
It is a p.d.f.,
f(x) = 2 = 0
1/2
0 ≤x≤1/2
otherwise
since f(x) > 0 for x = [0, 1/2] and [j f(x) dx = 1.
0
Example : Verify which of the following functions are p.d.f.s
(i)
f(x) = 3x2
0 ≤x≤1
= 0
; otherwise
(ii)
f(x) = 2 ex
; x20
= 0
; otherwise
Solution : To verity whether f (x) is p.d.f. we need to verify the
following two conditions (a) f (x) > 0x and (b) ( f (x) dx = 1,
00
(i) f (x) = 3x2 2 0x and j 3x2 dx = 3x2 dx = [x3] = 1
Therefore, f (x) is a p.d.f.
(ii)
f(x) = 2 ex 01X
00
0
[
=21
and 2 e-Xdx = 20 e-Xdx = 2 0
-oo
Distribution Function (D.F.) or Cumulative Distribution Function (C.D.F.)
A distribution function is defined in case of a continuous r.v. analogus. to that of discrete r.v. The summation is to be replaced by integration. Distribution function is an important entity in the field of statistical inference, reliability theory and life testing etc.
Definition: Let X be a continuous r.v. with p.d.f. f (x). The distribution function or cumulative distribution function denoted by F (x) is defined
as,
F(x) = P(X ≤ x)
;
- ∞ < x < ∞
=
( f (t) dt
Note: A r.v. X is defined to be continuous if F (x) is continuous.
Example 1.4: A r.v. X has p.d.f.
f(x) = 2 e 2x ; x > 0
= 0
Find its distribution function.
Solution: By definition,
otherwise
F(x) = f() dt = 2e2t dt = 2
= 1-e-2x
Since, F (x) is defined for all x e R we have to write,
F(x) = 0
= 1-e-2x ;
; x < 0 x 20
Note: (1) We have seen above how to find d.f. given the p.d.f. We can find p.d.f. from d.f. as follows:
f(x) =
F(x)
(2) Given the d.f. F (x) we can find P (a < X ≤ b) as follows: P(X ≤ b) P(X ≤ a)
p (a < x ≤ b) =
F (b) - F (a)
Properties of Distribution Function:
(1) Non-negative : F(x) is non-negative function.
That F(x) > 0, X
Proof : F(x) = P(X ≤ x) 20
F (x) being a probability it is always non-negative.
(2) Non-decreasing : F(x) is non-decreasing.
That is if a <b then, F (a) < F (b).
Proof : Let, A = {X | x ≤ a} and B = {X | x ≤ b} clearly A B
P(A) = P(B)
•
P(X ≤ a) s P (X ≤ b)
..
F (a) ≤ F (b).
lim
(3)
F (-∞0) =
F(x) = 0
lim
and F (0°) =
F(x)= 1.
Xxx
Note that, F (-∞) = P(X ≤∞) = 0, since {x | x <- } is impossible event. Similarly, F (∞) = P(X ≤ ∞) = 1, since {x | x < ∞} is a sure event.
(4) Continuity: F(x) is continuous for a continuous r.v. X. Proof of this property is beyond scope of the book.
Graph of F (x) of a continuous r.v. is a smooth continuous curve of the following type.
The above stated properties of distribution function are called as characteristic properties. It means every distribution function must satisfy these properties and any function satisfying these properties is a distribution function of some r.v.
F(x)
1.0-
0.75-
0.5-
0.25
Expectation of Continuous Random Variable
In order to find mean, variance, moments, M.G.F. etc. we need to learn the concept of expectation of a r.v.
We have seen how to find expectation of a discrete r.v. On similar lines it is defined for continuous r.v. The summation sign used in case of discrete r.v. is replaced by integration in case of continuous r.v.
Definition: Let X be a continuous r.v. with p.d.f. f (x), then the expectation or mathematical expectation of r.v. X is denoted by, E (X) and it is defined as,
E (X) =
x f (x) dx
00
provided the integral exists. (i.e. xl. f (x) dx < ∞).
Note: If g (x) is a real valued function of r.v. x. Then E [g (x)] is given
by,
E [g (x)] = g(x) f (x) dx.
Provided
Ig(x) f(x) dx < ∞o.
It is to be noted here, to find E [g (x)] we need not find p.d.f. of g (x), using p.d.f. of X we can find E [g (x)].
Example: Suppose, the life of a electronic component in hours
is a continuous r.v. X with p.d.f.
f(x) =
20000 x3
ix≥ 100.
Find the expected life in hours of the electronic component.
Theorems on Expectation:
Theorem 1: If c is a constant then E (c) = c.
Proof :
E (c) = cf (x) dx = c f (x) dx = c.
Theorem 2: Effect of change of origin and scale on E (X).
For a and b being constants.
E (X + b) = E(X) + b
E (aX) = a E (X)
(1)
(ii)
(iii)
E (a X + b)
a E (X) + b
Proof : (i) E (X + b) = (x + b) f (x) dx
( x f (x) dx + b ( f (x) dx = E (X) + b
00
(ii)
E (a X) =
0
ax f (x) dx = a
() x f (x) dx = a E (X)
(iii)
E (ax + b) = (ax + b) f (x) dx
00
x f (x) dx +
+bQ
f(x) dx
Note: E
= a E (X) + b E(X)-c d
for c and constants. We state two more
theorems, regarding bivariate r.v. which will be proved later on.
Theorem 3: If X and
are any two continuous r.v. s. then,
(i) E(X + Y) = E(X) +E (Y)
(ii) E (ax + by + c) = aE (X) + bE (Y) + c
Theorem 4: If X and Y are independent random variables then,
E (XY) =E(X). E(Y)
Moments of Continuous R.V.
(a) Raw moments: If X is a continuous r.v. with p.d.f. f (x) then rth
raw moment is denoted by u', and it is defined as,
H = E(X) = (J) x' f (x) dx
provided, x f (x) dx < ∞,
Therefore, the first four raw moments will be,
My
00
0
= x f (x) dx = mean, μ2 =
00
x2 f (x) dx
H = x3 f (x) dx and μ1 = x4 f (x) dx.
(b) Central moments: If E (X) = m then rth central moment is given
by,
Hr E(X-m)' =
(x - m)' f (x) dx
provided, (x-m)'| f (x) dx < ∞
Clearly,
M1 E(X-m) = 0, μ2 = E (X-m)2 = Var (X) μ3= E (X-m)3 and μ4= E (X-m)4
Relation between central moments and raw moments
μ. = 0
-(14)
M3 = μ2-3121 +2(1)3 сці том (ні) 2-3 (ці)
(c) Moments about 'a' will be
00
Hr (a) = E (x-a) = (x-a) f(x) dx, r = 1, 2, 3, ...
If particular for a = 0 we get, raw moments and a = E (X) we get
central moment.
Pearsonian coefficients :
μ3
B13Y1VB1 H3/"
H2
μ4
B22 12 B2-3.
Harmonic mean (H) is given by,
με
00
14
=
E() = 0) = f (x) dx
If X takes positive values only then, H is defined.
Geometric mean (G) is given by,
G = Antilog E (log X) = Antilog log x f (x) dx
(where, X is a r.v. taking non-negative values.)
Mean Deviation (M.D.) about mean (m) is given by
0
M.D. = E [|X - m❘] =
x-m❘ f (x) dx
Moment Generating Function (M.G.F.)
M.G.F. is useful in many ways in the study of probability distribution
such as,
(i) to find moments.
(ii) to find probability distribution of g(x); a function of r.v. X.
(iii) to verify whether the random variables are independent in case
of multivariate probability distribution.
Definition: If X is a continuous r.v. with p.d.f. f (x) then moment generating function of X is denoted by My (t) and defined as,
My (t) E (etx) etx f (x) dx
provided the integral exists for some h such that - h<t<h, h > 0.
Note: My (t) can be expressed as power series in t as follows:
I
t2 x2
My (t) = E (etx) = E (1 + tx +
+ ...)
21
+3
My (t)=1+μt+μ2·21+ H3.3!..
Raw moments using M.G.F.
urth raw moment
Method 1:
= coefficient of in the expansion of My (t).
Method 2:
H =
dr Mx (t)]
dt
t=0
Generating function for central moments:
If E (X) = m then M.G.F. for central moment will be,
Mx-m (t) = E [et (x-m)] = e-tm E (etx) = e- tm My (t)
It gives rth central moment μ, as follows,
Hr = coefficient of in the expansion of Mx-m (t).
Alternatively, μr =
dr Mx-m (t)
dt'
t = 0
Note: M.G.F. for moments about 'a' can be defined similarly. It is
Mx-a (t) eatMy (t).
Properties of M.G.F.
(1) Mx (0) = 1
(2) Effect of change of origin and scale.
Result : If X is a r.v. with M.G.F. Mx (t)
then,
(i) Mx + a(t) = Mx (t). eat
(ii) Mcx (t) = Mx (ct)
(iii) Ma + cx (t) = eat Mx (ct), where a, c are constants.
Proof : (i) Mx + a (t) =
E [e(x+a) t] = E [etx. eat] = eat Mx (t)
(ii) Mcx (t) =
E [ecx.t] = E [ex (ct)] = My (ct)
(iii) Ma + cx (t)
=
E [e(a + cx) t] = E [eat + cxt]
=
eat E [ex (ct)] = eat Mx (ct)
(3) If X and Y are independent r.v.s. then, Mxy (t)
Mx (t) - My (t)
Proof :
Mx + y (t) =
E [et (x + y)]
= E (etx.ety)
(: X and Y are independent)
= E (etx). E (ety)
= My (t). My (t).
Note: Max+by+c (t) = ect My (at) My (bt).
(4) Uniqueness property: For a given p.d.f. there is unique M.G.F. and for a given M.G.F. there is unique p.d.f.
Particularly, this result is useful in finding the distribution of transformation of random variables. If M.G.F. of a transformation coincides with that of standard probability distribution using this property we can conclude that the transformation follows the particular standard probability distribution. Thus, M.G.F. can be used to identify the probability distribution.
Cumulant Generating Function (C.G.F)
In order to obtain M.G.F. we have used the transformation etx, what do we get, if we use inverse transformation loge may be a question of interest. Accordingly, loge Mx (t) is obtained. It is called as cumulant generating function, which is found to be useful to find the central moments easily.
Definition : If X is a r.v. with M.G.F. My (t) then loge Mx (t) is called as cumulant generating function. It is denoted by Kx (t).
Thus, Kx (t) = loge Mx (t). Like M.G.F., Kx (t) can also be expressed as
a power series in t.
Kx (t) = k1t + k2 (t2/2!) + k3 (t3/3!) + ...
kr=rth cumulant = Coefficient of
of
in the expansion of Kx (t).
We can also get kr by successive differentiation of Kx (t).
d' Kx (t), kr =
dt Jt = 0
We summerise below the relation between cumulants and moments.
K1 = H, K2 = H2, K3 = H3, K1 = M1-3 μ2
Thus, we get first four central moments of X as follows.
H1 = 0, μ2 = K2, M3 = K3, M1 = K1 + 3 K2.
Median and Quartiles
If X is a continuous r.v. with p.d.f. f (x) then median is a point which divides the area under the curve y = f (x) into two equal parts. Therefore, median (u) is a value of r.v. X. such that
P(X ≤ μ) = P(X ≥ μ) = Ž
00
( f (x) dx = (J) f(x) dx =
μ
Mode
Definition: Mode is a value of X at which f (x) is maximum.
In case of a continuous random variable if p.d.f. f (x) is a differentiable function, then mode is that value of x which satisfies the
following conditions,
df (x) dx
= f(x) = and
d2f (x) dx2
= f(x)
< 0.
x = a
Symmetry of Continuous r.v.
A continuous r.v. X with p.d.f. f(x) is said to be symmetric around 'a' if f(ax) = f(a + x) Vx
Illustration: A r.v. X has p.d.f. f(x) =
1
()
- ∞ < x < ∞ is
symmetric around 5.
2
1
ーラ
Note that:
f(5-x) =
2√2
1
2√2π
(
()
f(5 + x) =
2√2π
..
f(5x) f(5+ x)
VXE (-∞∞, ∞0)
Note:
(1) If X is symmetric r.v. around 'a' then the graph of its p.d.f. is symmetric around 'a' like following figures.
f(x)
f(x)
f(x)
a
f(x)
Fig. 1.7
(2) Departure from symmetry can be measured using usual measures
of skewness such as:
(a) Karl Pearson's coefficient of skewness.
(b) Bowley's coefficient of skewness.
(c) Pearsonion coefficient of skewness ẞ1
(3) If r.v. X is symmetric then
i.e.
F(ax) = 1- F(a + x)
f(x) dx = f(x) dx
a + x
VXE (-∞∞, ∞∞)
(4) If X is symmetric around 0 then the M.G.F., Mx (t) is even function of t.
Transformations of Random Variable
Sometimes, we are interested in the probability distribution of function of a random variable. For example, given the probability distribution of X we may be interested in the distribution of ax + b, X2, IX, ex, log X etc. It may be useful in finding interrelationship between two distributions. Mainly, there are three methods of obtaining probability distribution of, function of r.v. as listed below:
(i) Using transformation method, which is to be used for strictly monotonic and one-to-one on to function.
(ii) Using distribution function.
(iii) Using M.G.F. of function of r.v.
According to the type of function the method is used.
Method (1) Suppose, X is a continuous r.v. with p.d.f. f (x) and distribution function F (x). Suppose, Y = g(x) is a strictly monotonic one-to-one, onto function. Then p.d.f. of Y is given by,
h (y) = f(x).
dx dy
Since, Y = g(x) is a strictly monotonic onto function, X = g1 (Y) is well defined hence,
h (y) = f [g-1 (Y)]
dx dy