Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If you need an accessible version of this item please contact JSTOR User Support

Random Variables with Independent Binary Digits

George Marsaglia
The Annals of Mathematical Statistics
Vol. 42, No. 6 (Dec., 1971), pp. 1922-1929
Stable URL: http://www.jstor.org/stable/2240118
Page Count: 8
  • Read Online (Free)
  • Download ($19.00)
  • Subscribe ($19.50)
  • Cite this Item
If you need an accessible version of this item please contact JSTOR User Support
Random Variables with Independent Binary Digits
Preview not available

Abstract

Let X = · b1b 2b 3 ⋯ be a random variable with independent binary digits bn taking values 0 or 1 with probability pn and qn = 1 - pn. When does X have a density? A continuous density? A singular distribution? This note gives necessary and sufficient conditions for the distribution of X to be: discrete: $\Sigma\min (p_n, q_n) < \infty$; singular: Σ∞ m[log (pn/qn)]2 = ∞ for every m; absolutely continuous: $\Sigma^\infty_m\lbrack\log (p_n/q_n)\rbrack^2 < \infty$ for some m. Furthermore, X has a density that is bounded away from zero on some interval if and only if log (pn/qn) is a geometric sequence with ratio 1/2 for $n > k$, and in that case the fractional part of 2k X has an exponential density (increasing or decreasing with the uniform a special case).

Page Thumbnails

  • Thumbnail: Page 
1922
    1922
  • Thumbnail: Page 
1923
    1923
  • Thumbnail: Page 
1924
    1924
  • Thumbnail: Page 
1925
    1925
  • Thumbnail: Page 
1926
    1926
  • Thumbnail: Page 
1927
    1927
  • Thumbnail: Page 
1928
    1928
  • Thumbnail: Page 
1929
    1929