On the linear combination of exponential and gamma random variables. (English) Zbl 1135.62307
Summary: The exact distribution of the linear combination \(\alpha X+\beta Y\) is derived when \(X\) and \(Y\) are exponential and gamma random variables distributed independently of each other. A measure of entropy of the linear combination is investigated. We also provide computer programs for generating tabulations of the percentage points associated with the linear combination. The work is motivated by examples in automation, control, fuzzy sets, neurocomputing and other areas of computer science.
Keywords:
entropy; exponential distribution; gamma distribution; linear combinations of random variables.Software:
RReferences:
| [1] | DOI: 10.1109/9.920787 · Zbl 1017.90052 · doi:10.1109/9.920787 |
| [2] | Cerruti, Counting the number of solutions of congruences, Application of Fibonacci Numbers vol 5 pp 85– (1993) · Zbl 0807.11004 |
| [3] | DOI: 10.1016/S0925-2312(98)00019-8 · Zbl 0908.68135 · doi:10.1016/S0925-2312(98)00019-8 |
| [4] | DOI: 10.1016/0165-0114(87)90031-5 · Zbl 0629.60006 · doi:10.1016/0165-0114(87)90031-5 |
| [5] | DOI: 10.1016/0165-0114(91)90157-L · Zbl 0722.60005 · doi:10.1016/0165-0114(91)90157-L |
| [6] | DOI: 10.1016/0165-0114(94)00175-7 · Zbl 0845.60006 · doi:10.1016/0165-0114(94)00175-7 |
| [7] | DOI: 10.1016/S0165-0114(97)00015-8 · Zbl 0941.60058 · doi:10.1016/S0165-0114(97)00015-8 |
| [8] | DOI: 10.1016/S0165-0114(00)00041-5 · Zbl 0994.60036 · doi:10.1016/S0165-0114(00)00041-5 |
| [9] | Feng, Note on: ”Sums of independent fuzzy random variables” [Fuzzy Sets and Systems2001, 123, 11–18], Fuzzy Sets and Systems 143 pp 479– (2004) · Zbl 1043.60024 · doi:10.1016/S0165-0114(03)00137-4 |
| [10] | DOI: 10.1111/j.1469-1809.1935.tb02120.x · doi:10.1111/j.1469-1809.1935.tb02120.x |
| [11] | DOI: 10.1214/aoms/1177729755 · Zbl 0040.36602 · doi:10.1214/aoms/1177729755 |
| [12] | DOI: 10.1214/aos/1176344729 · Zbl 0414.60033 · doi:10.1214/aos/1176344729 |
| [13] | DOI: 10.2307/2346911 · Zbl 0473.62025 · doi:10.2307/2346911 |
| [14] | DOI: 10.2307/2347721 · doi:10.2307/2347721 |
| [15] | DOI: 10.1080/03610928208828321 · Zbl 0511.62023 · doi:10.1080/03610928208828321 |
| [16] | DOI: 10.1007/BF02481123 · Zbl 0587.60015 · doi:10.1007/BF02481123 |
| [17] | DOI: 10.1080/03610928908802211 · Zbl 0685.62020 · doi:10.1080/03610928908802211 |
| [18] | DOI: 10.1002/sim.4780100317 · doi:10.1002/sim.4780100317 |
| [19] | DOI: 10.1080/03610929308831114 · Zbl 0784.62024 · doi:10.1080/03610929308831114 |
| [20] | DOI: 10.1109/24.285114 · doi:10.1109/24.285114 |
| [21] | DOI: 10.1080/00949659508811688 · Zbl 0842.62008 · doi:10.1080/00949659508811688 |
| [22] | DOI: 10.2307/1559029 · Zbl 1031.60007 · doi:10.2307/1559029 |
| [23] | Hitezenko, A note on a distribution of weighted sums of iid Rayleigh random variables, Sankhyā, A 60 pp 171– (1998) |
| [24] | DOI: 10.1016/S0378-3758(00)00149-X · Zbl 0974.60004 · doi:10.1016/S0378-3758(00)00149-X |
| [25] | Witkovský, Computing the distribution of a linear combination of inverted gamma variables, Kybernetika 37 pp 79– (2001) · Zbl 1263.62022 |
| [26] | Prudnikov, Integrals and Series (volumes 1, 2 and 3) (1986) |
| [27] | Gradshteyn, Table of Integrals, Series, and Products (2000) · Zbl 0981.65001 |
| [28] | DOI: 10.1002/j.1538-7305.1948.tb01338.x · Zbl 1154.94303 · doi:10.1002/j.1538-7305.1948.tb01338.x |
| [29] | Ihaka, R: A language for data analysis and graphics, Journal of Computational and Graphical Statistics 5 pp 299– (1996) |
| [30] | Rényi, On measures of entropy and information, Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability Vol. I pp 547– (1961) |
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