Truncated Distributions

Truncated distribution.

To understand the concept of truncated distribution, find the mean, variance of truncated distributions. To compute probabilities of events based on truncated distribution.

Introduction

Sometimes the usual random variables do not take all the values over its range. Some values are not taken by the random variable. Generally, some values in the beginning of range of random variable are discarded, then we say that r.v. has probability distribution truncated to the left. On the other hand, some values at the end are discarded; the r.v. is said to have a probability distribution truncated to the right. There may be a situation that the r.v. has distribution truncated to both the ends right and left. Thus a truncated probability distribution is a probability distribution observed on the reduced range. The probability distribution of such random variables obtained by increasing the respective probabilities proportionately so that the add up to unity. Such a modified probability mass function is the probability distribution of truncated distribution.

Probability Mass Function of Truncated Binomial Distribution

Definition: A binomial random variable X is said to be truncated to the left at X = 0 if it does not take value X = 0.

It is also called as a r.v. truncated below X = 0. In this case it takes values 1, 2,..., n.

Illustrations:

(1) Numbers of members (X) in family, it is a r.v. truncated to the left

of 0, since X takes values 1, 2, ..., n. The value 0 is discarded.

(2) Number of children (X) to the parents of an individual is a r.v. taking values 1, 2, 3, ..., n. Thus it is a truncated r.v. truncated below 0.

(3) The number of seeds (X) in pod. If is also a r.v. taking values 1, 2, ..., n. Thus it is a random variable truncated at X = 0.

(4) Suppose X is the number of seed geminated out 100 seeds from a packet. It is quite possible that X may not take value 0. Since it is rarely observed that no seed is germinated. Similarly, it may not be possible that all the seed (i.e. X = 100) will be germinated. Thus X will not take some values at both the ends. In this case X is r.v. truncated at below and above.

We derive the probability mass function (p.m.f.) of a binomial x.v. X truncated below at X = 0.

Suppose we denote the truncated r.v. by X, and the probability that the r.v. X, takes value x by P (X = x), where x = 1, 2, ..., n.

Note that: P (XT = x) =

P(X = X|X=0)

P (X = X, X = 0)

;

x = 0, 1, 2, ..., n

P (X = 0)

P (X = x)

;

x = 1, 2, ..., n.

1-P (X = 0)

"Cyp* q"-x

= 1-"Co p° q"

;

X→ B (n, p)

"Cx p* q^-* 1-q"

x = 1, 2, ..., n

Thus, the p.d.f. of binomial r.v. X truncated at X = 0 is given by

P (XT = x) =

"C, p* q"-x 1-q

= 0

Note : P (XT = x) > P (X = x).

x = 1, 2, ..., n otherwise

Mean and Variance of Truncated B (n, p)

Suppose X, follows binomial distribution truncated below at X = 0.

Σ Χ ( = x) = Σ x (X = x)

E (XT) =

x = 1

ΣΧ

x = 1

"C, p* q"-x

1-q"

x = 1

x "C, p* q"-x

Σ x. "Cx p*q" -x

(term x = 0 is 0)

x = 0

E (X)

X→ B (n, p)

E (XT) =

np 1-q"

Note:

(1) E (XT) =np/(1-q)

> np (1-q< 1). Thus the mean of truncated

r.v. (at X = 0) increases.

(2) As n increases E (XT) decreases upto np.

To find Var (XT), we find E [XT (XT-1)]

Note that,

E [XT (XT-1)] =

Σ x(x-1) P (XT = x)

x = 1

Σ x(x-1).

x = 0

"C. p* q^-* 1-q

(terms w.r.t. x = 0, 1, are 0)

1-q"

x = 0

"C, p* q-x 1-q

("terms w.r.t. x = 0, 1 are 0)

Σ x (x-1)

x = 2

x-(x-1) 1-1-2-2 p* q

n n- x

n(n-1) p2C-2 p*-2 q* =

1-q"

x = 2

E [XT (XT-1)] =

The second movement of XT is

n (n-1) p2

(q + p)^-2

1-q"

n (n-1) p2 1-q"

H= E(X) = E [X- -XT+

+X+]

= E [XT (XT - 1) + XT]

= E [XT (XT-1)] + E (X1)

n (n-1) p2

קח

1-q

1-q"

The variance of X, is

n (n-1) p2 1-q'

1-q

(n - 1) p + 1

1-q

(np - p + 1) (1 − q) – np

1-q"L

1-q

пр

[np-p+ 1- npq" + pq" - q" - np]

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(1-q)2

(1-q2 [q-npq" + pq" - q"]

(1-q

[q-q" (np-p+ 1)]

[q-q" (np + q)]

(1-q"

npq (1-q)

[1-q' (np + q)]

[1-q (np + q)]

= npq (1-q")

Note:

(1) Var (X,) = npq

(1 − q′′-1 (np + q)) (1-q)2

< npq.

Find:

(2) For large n Var (X+) = npq.

P.m.f. of Truncated Poisson (lambda)

We come across a situation of Poisson r.v. which does not take all values.

For example:

(1) Number of soldiers killed (X) during war. It may not be 0 or it may not take few smaller values.

(2) Number of telephone calls (X) received at a busy telephone line during peak hours. Certainly it will not take value 0 or few smaller values.

(3) Number of defective items (X) found in inspection of a good lot of large size. If the lot is good large values of X will not be observed.

(4) Number of pests or insects killed (X) due insecticide. X will not take value 0 or some lower values.

In the above situations the probability distribution is still Poisson distribution, however it is on reduced range. In order to have the total probability equal to 1, we increase each probability proportionately. If Σ p (x) = c, then we take probability of truncated r.v. XT = x as p (x)/c. P.m.f. of a Poisson r.v. truncated below at X = 0:

Suppose X → Poisson (2), which does not take value X = 0, we call it as a truncated distribution below at X = 0 or truncated distributed to the left at X 0. Thus, the value X = 0 is discarded. We denote X, as a r.v. truncated below at X = 0.

The p.m.f. of X, can be derived as follows:

P (X = x) = P(X = X|X = 0)

x = 0, 1, 2, ...

P (X = XnX0)

P (X = 0)

P (X = x)

P (X = 0)

ex/x!

x = 1, 2, ......

P(X = x) = 1-e/0!=

x! (1-e) x = 1, 2,......

p.m.f. of truncated Poisson r.v. below X = 0 is given by

e* 2*

P (X = x) =

x! (1-e)

= 0

x = 1, 2, ..., λ> 0

otherwise

Note:

(1)P (X = x) > P (X = x).

(ii) If X is truncated below at X = a (i.e. X does not take values 0, 1, 2,..., a 1, a) then the p.m.f. can be obtained similarly. It will be P(XX) = P(X = X|X > a)

P (X = X|X > a)

P (X > a)

x = a + 1, a + 2,.......

P (X = x)

1-P(X ≤ a)

eλ/x!

1-Σ (ex/x!)

x = a + 1, a + 2,......

(iii) If X is a r.v. truncated above X = a (i.e. X does not take values a,

a + 1, a +2, ...) the p.m.f. is

e λ/x!

P (X = x) = a