On the Distributions of the Product and the Quotient of the Independent and Uniformly Distributed Random Variables

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On the Distributions of the Product and the Quotient of the Independent and Uniformly Distributed

Random Variables.

by

H. SAKAMOTO, Tokyo.

1. Introduction. 'The object of this paper is to research the distributions of the product and the quotient of the independent

random variables, distributions of which are given by the probability functions, .

(1)

where Ar, αr,Ar-αr>0.Without loss of generality we assume here Ar =1,αr=εr,0<ε1<1,so that

(1')

w here 0<εr<1. A great number of papers about the distribution

of the. quotient of the random variables have been published. Among these, especially J. F. Steffensen has found the distribution of the quotient of the semi-normally distributed random variables(1 ). K. Pearson, M. Greenwood, C. Craig, S. D. Wicksell, T. Kaw ata and others have found the approximate value of the mean and variance and higher moments of the quotient of the random variables with the distributions of more generalized form.

In this paper I shall discuss the distribution functions of tii product and the quotient of random variables uniformly distributed in an interval.

Recently, Dr. Toshio Uno has found the distribution function of the sum of such random variables by using a the F o ii r i e r trans-.

form. .

(1) See Steffensen: On the semi-normal distribution, Skandinavisk Aktua-rietidskrift, 1937, p. 60.

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244 H. SAKAMOTO:

Following his method, we can also find the distribution functions of the product and the quotient of such random variables, but the process is more complicated in our case.

2. The distribution of a product of random variables. Let X1, X2, ...., Xn be mutually independent random variables and lot the distributions of Xr (r=1, 2, ...., n) be given by the proba-bility functions (1').

At first we shall find the distribution of the product X=X1X2

Xn. By taking the logarithm of X, we have

and we shall find the distribution of Z=log X.

Since Xi, X2, ...., Xn are mutually independent random variables, Z1, Z2, .... , Zn, Zr being log Xr, are also mutually independent random variables(1).

From the relation (1'), we can easily find the distribution func-tions of Xr, viz.

(2)

Thus the distribution functions of Zr are given by

(3)

where ar =log (1-ƒÀr), br = log (1+ƒÀr).

Consequently the characteristic function of Zr are calculated as

follows

(4)

Since Z1, Z2, ...., Zn are mutually independent, the characteristic

function of Z is Hence, by Levy's inversion formula (1),

we obtain the distribution function Fz(x) of Z as

(1) See, for example, C ramer: Random variables and probability distrbution. Kolmogoroff: Grundbegriffe der Wahrscheinlickheiterecheung.

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ON THE DISTRIBUTIONS OF THE PRODUCT ETC. 245

(5)

If dt are uniformly convergent for all

values-of x. thus we obtain

(6) by differentiating both sides of (5) with respect to x. Now, for the-sake of convenience, we put

(7)

which is easily verified to exist for m>=1. Then we have the following lemma.

Lemma.

(8) Here Eh denotes the symbolic operator(1), such that

Proof. Multiplying both sides of (7) by e-x, we get

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Obviously, dt is uniformly convergent for alb

values of x, in the case of m≧2. Hence, we obtain the relation

(10) by integrating both sides of (9) with respect to x over (0, x).

But the relation (10) also holds even in the case of m= 1. For

And the inner integral on the right is bounded for all T ; for it is equal to

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246 H. SAKAMOTO :

and

< a constant K2

for all valuos ofξand T,so that holds for all

values of T and x.

Hence, by Lebesgue's convergence theorem, we obtain

which is the relation (10) in the case of m=1. For other values of m we get

(5)

Then

(11)

Now

except the points x=a1, x=b1, where

Hence except x = a1, x = b1. Thus by

the lemma, we obtain

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Now the symbolic operator Eh and the integral operator are commutative, since

Therefore we obtain

(13)

Since M0 (x) dx=M1 (x) holds by the definitions of Mr (x), (13).

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248 H. SAKAMOTO:

(14)

Repeating such processes and considering the relations

we obtain

(15) Consequently the probability function of Z is given by

(16)

Here λ (E)denotes the operator

and may be considered as the polynomial of the operator E. Since

the relation holds symbolically. Hence (16)

may be written as follows

(17)

Consequently integrating both sides of(17) with respect to x over (-∞, x), we obtain

(18)

(7)

Therefore the distribution function of Z is given by

(19) Considerifig the relation Z=log X, we obtain

And

where▽denotes the symbolic operaor, such that

Hence, using this new operator, we obtain

(20)

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250 H, SAKAMOTO:

and noticing the relations

sue can find the distribution of X as

(21)

Now we can easily prove the following relations

(22) (23)

from the definition of the operator Δ. By the relation (22), we obtain

Hence the distribution of X is also given by the following probability

function

(24) 3. The mean value and the variance. The mean value E(X) and variance D2 (X) of such random variable are given by the follow-ing relations :

(9)

Hence

If er (r= 1, 2, . . , n) are sufficiently small,

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4. The distribution of a quotient of random variables. We shall find the distribution of the quotient of two

indepen-dent random variables X1, X2 in the similar way as in the case of a product.

Taking logarithm of X,

Putting

(28)

(29)

as the distribution functions of Z1, Z2.

Hence the characteristic functions of Z1, Z2 are calculated as follows,

(30) (31) Then the characteristic function of Z is given by

(32)

Consequently, by Levy's inversion formula, we get the distribition function of Z by the relation

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252 H. SAKAMOTO :

(33)

As dt are uniformly convergent for all values of x,

we get

(34)

which is the probability function of Z. Then in the quite similar way as in the case of a product, we get

Since

we obtain the relation

Thus,

(35)

whoreμ(E)denotes the operator

(11)

Hence

Here we obtain the relation

since

Consequently,

(36)

Integrating both sides of (36) with respect to x over(-∞,x),

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Considering the relation Z=log X, we obtain

(38)

where

Hence we find the distribution of X as

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254 T. SAKAMOTO :

By using the relation (22), we obtain

where

Consequently the distribution is also given by the following proba-bility function

(40)

5. The mean values and the variances. The mean value. E (X) and variance D2 (X) of such random variable are calculated as follows.

From the properties of f(x) the following relations always hold for sufficiently large values of M, viz.

(41)

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Hero we must evaluate the integrals

Considering the relation (23), we obtain

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(44) From (41) and (43), we get

(45)

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256 T. SAKAMOTO:

(46)

And here we have for sufficiently large value of M,

Consequently we obtain

(47)

(48)

Therefore, when ƒÃ1 ƒÃ2 are sufficiently small, we may. evaluate E (X), D2 (X) as follows.

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The result (47) and (48) will be also found from the distribution functions of X2 and

We can easily find the distribution of from the relation

(29),

(49)

Differentiating both sides of (49) with respect to x, we obtain

(50)

which is the probability function of And the mean values E (X) and D2(X) are calculated as follows.

If X1, X2 are mutually independent, then are mutually

inde-pendent(1). Hence the following relations holds.

(1) See, for example, Cramer: Random variables and probability distribution. Kolmo garoff: Grundbegrlffe der Wahrscheinlichkeitsrechnung..

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258 T. SAKAMOTO:

Thus the results (47) and (48) will be easily gotten.

6. Numerical values. In former sections we have found the distribution functions of the product X=X1X2.... Xn and the

quo-tient of the independent and uniformly distributed random

variables.

Here I will show graphically the behavior of the distributions of the sum, product and quotient of such random variables. Now,

we suppose that the random variables X1 and X2 has the probability functions.

(1) The probability function of the sum X=X1+X2 of two random variables.

(2) The probability function of the product X=X1X2 of two random variables.

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(3) The probability function of the quotiont of two

random variables.

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260 H. SAKAMOTO DISTRIBUTION OF THE PRODUCT ETC.

Fig. 2. The distributions of random variables, X=X1X2 and X=X2/X1.