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On the distribution of the time required to remove white balls from an urn
ON THE DISTRIBUTION OF THE TIME REQUIRED TO REMOVE WHITE BALLS FROM AN URN 8 MILTON SOBEL
Mathematics Department, Univeristy of California at Santa Barbara, Santa Barbara, CA 93106. U.S.A. 8 S. R. BERNARD
Health and Safety Research Division, Oak Ridge National Laboratory,* Oak Ridge, TN 37831, U.S.A. 8 V. R. R. UPPULURI
Mathematical Sciences Section, Engineering Physics and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A. Balls are removed one-at-a-time at equal time intervals from an urn initially containing w. white balls and a large number b of black balls and each black or white ball is immediately replaced by a black ball. The distribution of the number of white balls remaining after t iterations (under certain limiting operations) is taken from the literature. The problem is to use this result to find the time required to remo.ve a fixed number of white balls w1 from the urn. We then find the mean and variance of this distribution and also look at the special case when w1 = w@
I. Introduction.
Consider a model of an urn that has w. > 0 white balls at the outset and a large number of black balls b; in particular, b should be large relative to wc. Balls are removed in a small interval of time At and replaced by black ones (we assume that bAt + l/X, i.e. that b At e l/X). Since we are only removing white balls (not replacing them), this is a pure death process and we start with an expression for the probability that the number of white balls remaining after t time units is equal to k, namely
Pw,(k which appearsin
t) =
the literature
(i T
(e-Ar)k(l
(cf. e.g., Feller,
_
e-ht)~o-k,
1968, p. 498, or Parzen,
(1) 1961.
* Research sponsored by the Office of Health and Environmental Research, U.S. Department Energy under contract DE-ACOS-840R21400 with the Martin Marietta Energy Systems, Inc.
of
791
792
M. SOBEL et al.
p. 296). Our aim is to use (1) to derive the distribution of the time t required to remove any fixed number (or fixed proportion) of the white balls. Later we will attempt to consider some generalizations of this model by starting with several different colors to replace the one color white. 2. Derivation . To find the distribution of the time t = t(w) to remove w white balls, we break up the interval (0, t) into two disjoint parts (0, t - At) and (t - At, t) and use the same result (i) to remove w - 1 white balls in the first interval and (ii) to remove 1 white ball in the second interval, where w = w. - k since k is the number of units remaining and w. - k = w is the number removed. Since the intervals are disjoint, we simply multiply the probability of the first event by the conditional probability of the second event given the first. In this way, we obtain the density of the time g(t) required to remove a fixed number w of white balls as g(t)At =:P,o(w - 1, t-~t)P,v,,+,U, h(r-At) [em ~~-hAt
= Iyw,
1 wg-w
1 wg-w+l
[ 1 _
e-hAt]
[ 1 _
At> ,-A(r-At)
1 w-1
1
uwo+ 1) _ w + l~r(wfe~at)wO-w(
1 - e-at)w-le-hrhA t,
(2)
where terms of order (At)*, and higher, have been deleted. Replacing At by d r in the last expression of (2) gives the desired density, the rest of the expression in (2) is g(t). Let jr = 1 - e--ht so that if 0 < t < 7, then 0 < X < 1 - esAT. Clearly, X has the beta density, namely
uwo+ 1)
B(x) = rywo -
w +
w-y1 -x)WO-W* i)r(w)x
Hence the cdf (cumulative distribution and any fixed w < w. is
G(r)=
function) G(r) = P{t < T} for t = t(w)
1-e--h7 r(w,+ 1) xw-‘(1 r(w,- w + i)r(w) I0
= zl_e_hT(w’ wo - w +
where ZP(a, 6) is the usual incomplete 1934).
-X)WD-W;x
113 beta function
(4) (tabulated
by Pearson,
REMOVINGWHITEBALLSFROMANURN
793
We are particularly interested in the expectation and variance of the time required to remove a fixed number (say, wi) of the white balls under this model, where wi < we. Some of the results obtained below are similar to those obtained by Epstein and Sobel (1954); another reference for related results is the relation of Dirichlet C-integrals to harmonic series in Vol. 9 of Selected Tables of Mathematical Statistics (Sobel et al., 1985). THEOREM
1.The expected time required to remove w1 white balls is 1
Ht I Wl,
WI), A) =
w1-1
1
x -w,--j A
-
(5)
j=o
and the second moment given by
of the time required
to remove w1 white balls is
::+g3’(g5)(&)
E(t2 I WI, wo, A) = -
Proof. Putting t in terms of y = e-”
(so that 0 < y < 1) we have
I--two+ 1) x r(w, - wi + l)F(w,)
El) I WI, WI), h) = 1
C6)
’ s0
[log (l/y)1 y”o-“‘(1
-y)“‘-‘@
where integration by parts was used in (7). To get an equivalent form for (7) it is not difficult to see that we can write the result in (7) as a double integral '(1 o
-x)wl-~xwO-w~
1
(1 -
y)iywO-i
dy
w1-1
z-x hj=O
&
dy
1
wo-j'
(8)
In the first step of (8): we made use (in reverse) of the relation between the tail of a binomial sum and the incomplete beta integral; in the last step we inverted the order of summation and integration and used the complete beta integral. This completes the proof of the first result in Theorem 1. For the special case in which \ol = \vo we get the interesting result
M. SOBEL et al.
794
E(t I w,, wo, A) =
‘2
1 (9)
Xjzl7
namely l/X times the harmonic series up to w,-,. To prove (6), we start in the same way by putting [log (l/y)] 2/X2 and integrating by parts; this gives
for t2 the quantity
First we write this result (10) as a triple integral and then we use the first step in (8) to obtain 2
Uwof
1)
-x)w’-lxwo-wl~ dvck xZr(wo - v,l+ I~r~w,)16~~~;~~(1
(1 - y)iywo-i-l dy
(11)
Integrating by parts again, after expanding ( 1 - y)j, gives us
2
wt-1
(--1)O j wo j P a=0 (w. - j + c~)~ J
Xz
0
wO
(12) (y j
r(w,+ 1)
=$w$l+_ j
j=O
z(
cy
0
.
0
i
l),=o(~o
j r(wo-j)r(j+
7)
j+d2 (y
and we now use the final result between (7) and (8) above (for any wl) with IY~and (Yunchanged and wl - 1 in (7) replaced by j to obtain
E02I wo,WI, N =
;wgl--&$ 5, j0
0
i0
0
(13)
which is the same as the result given in (6) above. This completes the proof of Theorem 1.
REMOVINGWHITEBALLSFROMANURN
795
For the special case w1 = w. the result in (13) reduces to
and hence for this case [using (9)3 the variance reduces to the simple form I w. 1 2 - . 02(t I wg, wg, A) = A2i,l iI 4
(15)
Another possible method to derive these moments is to first obtain the moment generating functions q(s). This is easily shown to be g(s) = r[wo
r(w,+ 1) 1 + (S/h)] lyw,--ww, + 1) r[w,+ 1 + (S/X)]’ - Wl +
(16)
However, this method involves the differentiation of gamma functions, which we prefer to avoid., Let Tf denote the time between the removal of the ith and (i - 1)th white ball, and define To = 0. Then the sum of the I;: (i = 1, 2, . . . , w) = T(w) denotes the total time necessary to remove w white balls. It is interesting to note that if we assume the Ti are successive differences of the first w order statistics from a set of w. independent negative exponentials with scale parameter h, then the expectation of q is [ X(w, - i + l>]-’ and the second moment is 2[ h(wo - i + 1)la2. Hence we obtain for T(w) = 5 rr;:, i=1 using the independence
in deriving (18)
E{T(w)) =
X2f=jiyFl
(17)
1)’
h(wo
i+j
0
22
g1 ‘i + P(w,--i+
l)(wg-j+
1
g (Woe
i + 1)(1110-j + 1)’
which for NJ = IZ’~are identical with the results obtained above. respectively.
1)
(18) in (9) and (13)
M. $OBEL et al.
796
3. Illustration
of the Use of These Results.
The results above can be example illustrates only one of these
used in various ways; our present possible ways. For X = 1 and w. = 1012, find the time r such that we can assert with co.nfidence 0.95 that at least 90% of the white balls are removed. Setting X = 1, we obtain from (4) the result 11--e_h7(0.9Wo,0.1 wa + 1) = 0.95
(19)
and we have to solve for r when w. = 10r2. Using the normal approximation to the beta distribution, we obtain
I
P z<
1 -e7-0.9 3( lo-‘)
I = 0.95 * 0.1 - e7 = 3( lo-‘)( 1.645) = 0.0000005,
(20)
where Z is a standard normal random variable. It follows easily that T = 2.3026 time units; the unit of time is the same as that used for X. It is interesting to note that we get the same estimate of r (though a nonprobabilistic one) from the simple wash-out model dw
W =
dt=-
Then units. As in the
(b
+
--xw,
w(0)
=
WI-J.
(21)
w,>At
w(t) = woe-“’ and solving e-*’ = 0.1 with h = 1 gives t = 2.3026 time
This may help to make our answer reasonable and more intuitive. indicated in (20) we have a correction to the above answer (21), but example above the correction was too small to effect the result.
4. Relevance of this Model to Biological Problems. On p. 74 of Part 1 of the report on Limits of Intakes of Radionuclides by Workers, prepared by the ICRP, a retention equation with four exponential terms is given. This expression gives the fractional amount of cobalt in a human body, when a unit amount is injected at time zero. This is based on the data of Smith et al. (1972) which led to the retention function R(t) = 0,5e-0*69s’/o.s+ 0.3e-O*@sr/6+ 0.1 e-O*693r/@+ O.le-O*693f/~.
(22)
We interpret this as a system consisting of four independent urns, where initially at t = 0, there is 50% of the input in the first urn, 30% in the second urn, 10% in the third and fourth urns. From equation (22), we see that after half a day, i.e. t = 1, the first term reduces to 0.5 e-o*693= 0.25, which corresponds to the dilution of white balls in the first urn by half. Similarly, after six days, i.e. t = 6, the second term in equation (22) reduces to 0.3 e-“.693 = 0.15 which corresponds to the dilution of white balls in the second urn by half.
REMOVING~ITCBALLSFROMANURN
791
From the first term in equation (22) we can interpret 0.693/0.5 to be the rate at which the white balls in urn 1 are being depleted. Similarly, from the second term in equation (22), we can interpret 0.693/6 to be the rate at which the white balls in urn 2 are being depleted. This depletion rate in an urn is denoted by h in equation (5). From equation (22), we also note that as t + 00,R(m) = 0, which corresponds to the depletion of all the white balls. In this paper, we discussed the average and variance of the waiting times required for the complete depletion in an urn. There are other sources of data pertaining to the retention of other elements which are referred to in the ICRP publication 30 (1979). The authors wish to thank Dr Mary Leitnaker of the University of Tennessee for pointing out in (17) and (18) the interesting analogs of our results.
LITERATURE Epstein, B. and M. Sobel. 1954. “Some Theorems Relevant to Life Testing from an Exponential Distribution.” Ann. Math. Statist. 25, 373-38 1. Feller, W. 1968. An Introduction to Probability Theory and its Applications, Vol. I, 3rd edn. New York: John Wiley. International Commission on Radiological Protection. 1979, ICRP Publication 30, Part I. Limits for Intakes of Radionuclides by Workers. Oxford: Pergamon Press. Parzen, E. 1962. Stochastic Processes. San Francisco: Holden-Day. Pearson, K. 1934. Tables of the Incomplere Beta-Functions. Cambridge, U.K.: Cambridge University Press. Sobel, M., V. R. R. Uppuluri and K. Frankowski. 1985. Selected Tables in Mathematical Statistics, Vol. 9, Dirichlet Distribution-Type 2. Providence, Rhode Island: The American Mathematical Society. Smith, T., C. J. Edmonds and C. F. Bamaby. “Absorption and Retention of Cobalt in Man by Wholebody Counting.” Health Phys. 22, 359-367.
l-28-85 REVISED 5-7.85
RECEIVED
Mathematics Department, Univeristy of California at Santa Barbara, Santa Barbara, CA 93106. U.S.A. 8 S. R. BERNARD
Health and Safety Research Division, Oak Ridge National Laboratory,* Oak Ridge, TN 37831, U.S.A. 8 V. R. R. UPPULURI
Mathematical Sciences Section, Engineering Physics and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A. Balls are removed one-at-a-time at equal time intervals from an urn initially containing w. white balls and a large number b of black balls and each black or white ball is immediately replaced by a black ball. The distribution of the number of white balls remaining after t iterations (under certain limiting operations) is taken from the literature. The problem is to use this result to find the time required to remo.ve a fixed number of white balls w1 from the urn. We then find the mean and variance of this distribution and also look at the special case when w1 = w@
I. Introduction.
Consider a model of an urn that has w. > 0 white balls at the outset and a large number of black balls b; in particular, b should be large relative to wc. Balls are removed in a small interval of time At and replaced by black ones (we assume that bAt + l/X, i.e. that b At e l/X). Since we are only removing white balls (not replacing them), this is a pure death process and we start with an expression for the probability that the number of white balls remaining after t time units is equal to k, namely
Pw,(k which appearsin
t) =
the literature
(i T
(e-Ar)k(l
(cf. e.g., Feller,
_
e-ht)~o-k,
1968, p. 498, or Parzen,
(1) 1961.
* Research sponsored by the Office of Health and Environmental Research, U.S. Department Energy under contract DE-ACOS-840R21400 with the Martin Marietta Energy Systems, Inc.
of
791
792
M. SOBEL et al.
p. 296). Our aim is to use (1) to derive the distribution of the time t required to remove any fixed number (or fixed proportion) of the white balls. Later we will attempt to consider some generalizations of this model by starting with several different colors to replace the one color white. 2. Derivation . To find the distribution of the time t = t(w) to remove w white balls, we break up the interval (0, t) into two disjoint parts (0, t - At) and (t - At, t) and use the same result (i) to remove w - 1 white balls in the first interval and (ii) to remove 1 white ball in the second interval, where w = w. - k since k is the number of units remaining and w. - k = w is the number removed. Since the intervals are disjoint, we simply multiply the probability of the first event by the conditional probability of the second event given the first. In this way, we obtain the density of the time g(t) required to remove a fixed number w of white balls as g(t)At =:P,o(w - 1, t-~t)P,v,,+,U, h(r-At) [em ~~-hAt
= Iyw,
1 wg-w
1 wg-w+l
[ 1 _
e-hAt]
[ 1 _
At> ,-A(r-At)
1 w-1
1
uwo+ 1) _ w + l~r(wfe~at)wO-w(
1 - e-at)w-le-hrhA t,
(2)
where terms of order (At)*, and higher, have been deleted. Replacing At by d r in the last expression of (2) gives the desired density, the rest of the expression in (2) is g(t). Let jr = 1 - e--ht so that if 0 < t < 7, then 0 < X < 1 - esAT. Clearly, X has the beta density, namely
uwo+ 1)
B(x) = rywo -
w +
w-y1 -x)WO-W* i)r(w)x
Hence the cdf (cumulative distribution and any fixed w < w. is
G(r)=
function) G(r) = P{t < T} for t = t(w)
1-e--h7 r(w,+ 1) xw-‘(1 r(w,- w + i)r(w) I0
= zl_e_hT(w’ wo - w +
where ZP(a, 6) is the usual incomplete 1934).
-X)WD-W;x
113 beta function
(4) (tabulated
by Pearson,
REMOVINGWHITEBALLSFROMANURN
793
We are particularly interested in the expectation and variance of the time required to remove a fixed number (say, wi) of the white balls under this model, where wi < we. Some of the results obtained below are similar to those obtained by Epstein and Sobel (1954); another reference for related results is the relation of Dirichlet C-integrals to harmonic series in Vol. 9 of Selected Tables of Mathematical Statistics (Sobel et al., 1985). THEOREM
1.The expected time required to remove w1 white balls is 1
Ht I Wl,
WI), A) =
w1-1
1
x -w,--j A
-
(5)
j=o
and the second moment given by
of the time required
to remove w1 white balls is
::+g3’(g5)(&)
E(t2 I WI, wo, A) = -
Proof. Putting t in terms of y = e-”
(so that 0 < y < 1) we have
I--two+ 1) x r(w, - wi + l)F(w,)
El) I WI, WI), h) = 1
C6)
’ s0
[log (l/y)1 y”o-“‘(1
-y)“‘-‘@
where integration by parts was used in (7). To get an equivalent form for (7) it is not difficult to see that we can write the result in (7) as a double integral '(1 o
-x)wl-~xwO-w~
1
(1 -
y)iywO-i
dy
w1-1
z-x hj=O
&
dy
1
wo-j'
(8)
In the first step of (8): we made use (in reverse) of the relation between the tail of a binomial sum and the incomplete beta integral; in the last step we inverted the order of summation and integration and used the complete beta integral. This completes the proof of the first result in Theorem 1. For the special case in which \ol = \vo we get the interesting result
M. SOBEL et al.
794
E(t I w,, wo, A) =
‘2
1 (9)
Xjzl7
namely l/X times the harmonic series up to w,-,. To prove (6), we start in the same way by putting [log (l/y)] 2/X2 and integrating by parts; this gives
for t2 the quantity
First we write this result (10) as a triple integral and then we use the first step in (8) to obtain 2
Uwof
1)
-x)w’-lxwo-wl~ dvck xZr(wo - v,l+ I~r~w,)16~~~;~~(1
(1 - y)iywo-i-l dy
(11)
Integrating by parts again, after expanding ( 1 - y)j, gives us
2
wt-1
(--1)O j wo j P a=0 (w. - j + c~)~ J
Xz
0
wO
(12) (y j
r(w,+ 1)
=$w$l+_ j
j=O
z(
cy
0
.
0
i
l),=o(~o
j r(wo-j)r(j+
7)
j+d2 (y
and we now use the final result between (7) and (8) above (for any wl) with IY~and (Yunchanged and wl - 1 in (7) replaced by j to obtain
E02I wo,WI, N =
;wgl--&$ 5, j0
0
i0
0
(13)
which is the same as the result given in (6) above. This completes the proof of Theorem 1.
REMOVINGWHITEBALLSFROMANURN
795
For the special case w1 = w. the result in (13) reduces to
and hence for this case [using (9)3 the variance reduces to the simple form I w. 1 2 - . 02(t I wg, wg, A) = A2i,l iI 4
(15)
Another possible method to derive these moments is to first obtain the moment generating functions q(s). This is easily shown to be g(s) = r[wo
r(w,+ 1) 1 + (S/h)] lyw,--ww, + 1) r[w,+ 1 + (S/X)]’ - Wl +
(16)
However, this method involves the differentiation of gamma functions, which we prefer to avoid., Let Tf denote the time between the removal of the ith and (i - 1)th white ball, and define To = 0. Then the sum of the I;: (i = 1, 2, . . . , w) = T(w) denotes the total time necessary to remove w white balls. It is interesting to note that if we assume the Ti are successive differences of the first w order statistics from a set of w. independent negative exponentials with scale parameter h, then the expectation of q is [ X(w, - i + l>]-’ and the second moment is 2[ h(wo - i + 1)la2. Hence we obtain for T(w) = 5 rr;:, i=1 using the independence
in deriving (18)
E{T(w)) =
X2f=jiyFl
(17)
1)’
h(wo
i+j
0
22
g1 ‘i + P(w,--i+
l)(wg-j+
1
g (Woe
i + 1)(1110-j + 1)’
which for NJ = IZ’~are identical with the results obtained above. respectively.
1)
(18) in (9) and (13)
M. $OBEL et al.
796
3. Illustration
of the Use of These Results.
The results above can be example illustrates only one of these
used in various ways; our present possible ways. For X = 1 and w. = 1012, find the time r such that we can assert with co.nfidence 0.95 that at least 90% of the white balls are removed. Setting X = 1, we obtain from (4) the result 11--e_h7(0.9Wo,0.1 wa + 1) = 0.95
(19)
and we have to solve for r when w. = 10r2. Using the normal approximation to the beta distribution, we obtain
I
P z<
1 -e7-0.9 3( lo-‘)
I = 0.95 * 0.1 - e7 = 3( lo-‘)( 1.645) = 0.0000005,
(20)
where Z is a standard normal random variable. It follows easily that T = 2.3026 time units; the unit of time is the same as that used for X. It is interesting to note that we get the same estimate of r (though a nonprobabilistic one) from the simple wash-out model dw
W =
dt=-
Then units. As in the
(b
+
--xw,
w(0)
=
WI-J.
(21)
w,>At
w(t) = woe-“’ and solving e-*’ = 0.1 with h = 1 gives t = 2.3026 time
This may help to make our answer reasonable and more intuitive. indicated in (20) we have a correction to the above answer (21), but example above the correction was too small to effect the result.
4. Relevance of this Model to Biological Problems. On p. 74 of Part 1 of the report on Limits of Intakes of Radionuclides by Workers, prepared by the ICRP, a retention equation with four exponential terms is given. This expression gives the fractional amount of cobalt in a human body, when a unit amount is injected at time zero. This is based on the data of Smith et al. (1972) which led to the retention function R(t) = 0,5e-0*69s’/o.s+ 0.3e-O*@sr/6+ 0.1 e-O*693r/@+ O.le-O*693f/~.
(22)
We interpret this as a system consisting of four independent urns, where initially at t = 0, there is 50% of the input in the first urn, 30% in the second urn, 10% in the third and fourth urns. From equation (22), we see that after half a day, i.e. t = 1, the first term reduces to 0.5 e-o*693= 0.25, which corresponds to the dilution of white balls in the first urn by half. Similarly, after six days, i.e. t = 6, the second term in equation (22) reduces to 0.3 e-“.693 = 0.15 which corresponds to the dilution of white balls in the second urn by half.
REMOVING~ITCBALLSFROMANURN
791
From the first term in equation (22) we can interpret 0.693/0.5 to be the rate at which the white balls in urn 1 are being depleted. Similarly, from the second term in equation (22), we can interpret 0.693/6 to be the rate at which the white balls in urn 2 are being depleted. This depletion rate in an urn is denoted by h in equation (5). From equation (22), we also note that as t + 00,R(m) = 0, which corresponds to the depletion of all the white balls. In this paper, we discussed the average and variance of the waiting times required for the complete depletion in an urn. There are other sources of data pertaining to the retention of other elements which are referred to in the ICRP publication 30 (1979). The authors wish to thank Dr Mary Leitnaker of the University of Tennessee for pointing out in (17) and (18) the interesting analogs of our results.
LITERATURE Epstein, B. and M. Sobel. 1954. “Some Theorems Relevant to Life Testing from an Exponential Distribution.” Ann. Math. Statist. 25, 373-38 1. Feller, W. 1968. An Introduction to Probability Theory and its Applications, Vol. I, 3rd edn. New York: John Wiley. International Commission on Radiological Protection. 1979, ICRP Publication 30, Part I. Limits for Intakes of Radionuclides by Workers. Oxford: Pergamon Press. Parzen, E. 1962. Stochastic Processes. San Francisco: Holden-Day. Pearson, K. 1934. Tables of the Incomplere Beta-Functions. Cambridge, U.K.: Cambridge University Press. Sobel, M., V. R. R. Uppuluri and K. Frankowski. 1985. Selected Tables in Mathematical Statistics, Vol. 9, Dirichlet Distribution-Type 2. Providence, Rhode Island: The American Mathematical Society. Smith, T., C. J. Edmonds and C. F. Bamaby. “Absorption and Retention of Cobalt in Man by Wholebody Counting.” Health Phys. 22, 359-367.
l-28-85 REVISED 5-7.85
RECEIVED