\[ X = \ln\left(\frac{1 - p}{1 - p^U}\right) = \ln(1 - p) - \ln\left(1 - p^U \right) \] When the minimum value of x equals 0, the equation reduces to this. \[ \lim_{p \to 1} G^c(x) = \lim_{p \to 1} \frac{p e^{-x}}{1 - (1 - p) e^{-x}} = e^{-x}, \quad x \in [0, \infty) \] The term ‘exponent’ implies the ‘power’ of a number. A generic term of the sequence has probability density function where is the support of the distribution and the rate parameter is the parameter that needs to be estimated. Describe the properties of graphs of exponential functions. If the base, [latex]b[/latex], is equal to [latex]1[/latex], then the function trivially becomes [latex]y=a[/latex]. The log of a base e is called the natural log of a … Machler2012. The standard exponential-logarithmic distribution has decreasing failure rate. \(\newcommand{\N}{\mathbb{N}}\) The points [latex](0,1)[/latex] and [latex](1,b)[/latex] are always on the graph of the function [latex]y=b^x[/latex]. If we instead consider logarithmic functions with a base [latex]b[/latex], such that [latex]0

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