On the Exact Distribution for the Product and Ratio of Normal and t Random Variables

2006 ◽  
Vol 58 (1-2) ◽  
pp. 77-92
Author(s):  
Saralees Nadarajah ◽  
B. M. Golam Kibria
Keyword(s):  
Random Variables ◽  
Exact Distribution

Sample Size Calculation for Poisson Endpoint Using the Exact Distribution of Difference Between Two Poisson Random Variables

2011 ◽  
Vol 3 (3) ◽  
pp. 497-504 ◽  
Author(s):  
Sandeep Menon ◽  
Joseph Massaro ◽  
Jerry Lewis ◽  
Michael Pencina ◽  
Yong-Cheng Wang ◽  
...  
Keyword(s):  
Sample Size ◽  
Sample Size Calculation ◽  
Random Variables ◽  
Exact Distribution

The log-normal approximation in financial and other computations

2004 ◽  
Vol 36 (03) ◽  
pp. 747-773 ◽  
Author(s):  
Daniel Dufresne
Keyword(s):  
Brownian Motion ◽  
Explicit Formula ◽  
Normal Approximation ◽  
Random Variables ◽  
Exact Distribution ◽  
Financial Mathematics ◽  
Asian Options ◽  
Basket Options ◽  
Asymptotic Formulae ◽  
Log Normal

Sums of log-normals frequently appear in a variety of situations, including engineering and financial mathematics. In particular, the pricing of Asian or basket options is directly related to finding the distributions of such sums. There is no general explicit formula for the distribution of sums of log-normal random variables. This paper looks at the limit distributions of sums of log-normal variables when the second parameter of the log-normals tends to zero or to infinity; in financial terms, this is equivalent to letting the volatility, or maturity, tend either to zero or to infinity. The limits obtained are either normal or log-normal, depending on the normalization chosen; the same applies to the reciprocal of the sums of log-normals. This justifies the log-normal approximation, much used in practice, and also gives an asymptotically exact distribution for averages of log-normals with a relatively small volatility; it has been noted that all the analytical pricing formulae for Asian options perform poorly for small volatilities. Asymptotic formulae are also found for the moments of the sums of log-normals. Results are given for both discrete and continuous averages. More explicit results are obtained in the case of the integral of geometric Brownian motion.

Exact Distribution for the Product of Two Correlated Gaussian Random Variables

2016 ◽  
Vol 23 (11) ◽  
pp. 1662-1666 ◽  
Author(s):  
Guolong Cui ◽  
Xianxiang Yu ◽  
Salvatore Iommelli ◽  
Lingjiang Kong
Keyword(s):  
Random Variables ◽  
Exact Distribution ◽  
Gaussian Random Variables

Some exact distributions of a last one-sided exit time in the simple random walk

1986 ◽  
Vol 23 (02) ◽  
pp. 332-340
Author(s):  
Chern-Ching Chao ◽  
John Slivka
Keyword(s):  
Random Walk ◽  
Positive Integer ◽  
Exit Time ◽  
Random Variables ◽  
Random Variable ◽  
Simple Random Walk ◽  
Exact Distribution ◽  
Partial Sum ◽  
Exact Distributions ◽  
Special Case

For each positive integer n, let Sn be the nth partial sum of a sequence of i.i.d. random variables which assume the values +1 and −1 with respective probabilities p and 1 – p, having mean μ= 2p − 1. The exact distribution of the random variable , where sup Ø= 0, is given for the case that λ > 0 and μ+ λ= k/(k + 2) for any non-negative integer k. Tables to the 99.99 percentile of some of these distributions, as well as a limiting distribution, are given for the special case of a symmetric simple random walk (p = 1/2).

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