The function [latex]y=b^x[/latex] takes on only positive values because any positive number [latex]b[/latex] will yield only positive values when raised to any power. If we instead consider logarithmic functions with a base [latex]b[/latex], such that [latex]00[/latex], the graph of [latex]y=log{_b}x[/latex] and the graph of [latex]y=log{_\frac{1}{b}}x[/latex] are symmetric over the [latex]x[/latex]-axis. \[ G^{-1}(u) = \ln\left(\frac{1 - p}{1 - p^{1 - u}}\right) = \ln(1 - p) - \ln\left(1 - p^{1 - u}\right), \quad u \in [0, 1) \]. Suppose also that \( N \) has the logarithmic distribution with parameter \( 1 - p \in (0, 1) \) and is independent of \( \bs T \). The point [latex](1,b)[/latex] is always on the graph of an exponential function of the form [latex]y=b^x[/latex] because any positive number [latex]b[/latex] raised to the first power yields [latex]1[/latex]. Doing so you can obtain the following points: [latex](-2,4)[/latex], [latex](-1,2)[/latex], [latex](0,1)[/latex], [latex](1,\frac{1}{2})[/latex] and [latex](2,\frac{1}{4})[/latex]. \[ X = \ln\left(\frac{1 - p}{1 - p^U}\right) = \ln(1 - p) - \ln\left(1 - p^U \right) \] Suppose that \( X \) has the exponential-logarithmic distribution with shape parameter \( p \in (0, 1) \) and scale parameter \( b \in (0, \infty) \). Hence Acidic or Alkaline Recall that \( F^{-1}(u) = b G^{-1}(u) \) where \( G^{-1} \) is the quantile function of the standard distribution. When only the [latex]y[/latex]-axis has a log scale, the exponential curve appears as a line and the linear and logarithmic curves both appear logarithmic.It should be noted that the examples in the graphs were meant to illustrate a point and that the functions graphed were not necessarily unwieldy on a linearly scales set of axes. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. Note that \( V_i = T_i / b \) has the standard exponential distribution. At first glance, the graph of the logarithmic function can easily be mistaken for that of the square root function. If \( U \) has the standard exponential distribution then Hence \( \E(X^n) = b^n \E(Z^n) \). The domain of the function is all positive numbers. Namely, [latex]y=log{_b}x[/latex]. Hence \( X = b Z \) has the exponential-logarithmic distribution with shape parameter \( p \) and scale parameter \( b \). Again, since the quantile function of the exponential-logarithmic distribution has a simple closed form, the distribution can be simulated using the random quantile method. \[ U = \frac{\ln\left[1 - (1 - p) e^{-X / b}\right]}{\ln(p)} \] Hence, using the polylogarithm of order 1 (the standard power series for the logarithm), The polylogarithm functions of orders 0, 1, 2, and 3. As \( p \downarrow 0 \), the numerator in the last expression for \( \E(X^n) \) converges to \( n! Why is this so? \frac{\Li_{n+1}(1 - p)}{\ln(p)}, \quad n \in \N \]. If \( U \) has the standard uniform distribution then For \( s \in \R \), Properties of the distribution Distribution That is, as [latex]x[/latex] approaches zero the graph approaches negative infinity. Some functions with rapidly changing shape are best plotted on a scale that increases exponentially, such as a logarithmic graph. Secondly, it allows one to interpolate at any point on the plot, regardless of the range of the graph. Recall that \( R(x) = \frac{1}{b} r\left(\frac{x}{b}\right) \) for \( x \in [0, \infty) \), where \( r \) is the failure rate function of the standard distribution. Hence \( U = 1 - G(X) \) also has the standard uniform distribution. To see this, think of an exponential random variable in the sense of tossing a lot of coins until observing the first heads. But then \( Y = c X = (b c) Z \). Thus, the log function is the inverse of exponentiation and has the following properties: In this website we use logs with base = 10 (called log base 10 and written simply as log a) and logs with base e where e is a special constant equal to 2.718282…. The failure rate function \( R \) of \( X \) is given by. \[ \E(X^n) = -\frac{1}{\ln(p)} \int_0^\infty \sum_{k=1}^\infty (1 - p)^k x^n e^{-k x} dx = -\frac{1}{\ln(p)} \sum_{k=1}^\infty (1 - p)^k \int_0^\infty x^n e^{-k x} dx \] \[ g(x) = -\frac{\Li_0\left[(1 - p) e^{-x}\right]}{\ln(p)} = \frac{\Li_0\left[(1 - p) e^{-x}\right]}{\Li_1(1 - p)}, \quad x \in [0, \infty) \] The distribution of \( Z \) converges to the standard exponential distribution as \( p \uparrow 1 \) and hence the the distribution of \( X \) converges to the exponential distribution with scale parameter \( b \). The function [latex]y=b^x[/latex] takes on only positive values and has the [latex]x[/latex]-axis as a horizontal asymptote. has the standard exponential-logarithmic distribution with shape parameter \( p \). (Assume that the time that elapses from one bus to the next has exponential distribution, which means the total number of buses to arrive during an hour has Poisson distribution.) To do so, we interchange [latex]x[/latex] and [latex]y[/latex]: The exponential function [latex]3^x=y[/latex] is one we can easily generate points for. Where a normal (linear) graph might have equal intervals going 1, 2, 3, 4, a logarithmic scale would have those same equal intervals represent 1, 10, 100, 1000. Before this point, the order is reversed. As you connect the points you will notice a smooth curve that crosses the y-axis at the point [latex](0,1)[/latex] and is decreasing as [latex]x[/latex] takes on larger and larger values. This means that the curve gets closer and closer to the [latex]y[/latex]-axis but does not cross it. As a function of \( x \), this is the reliability function of the exponential-logarithmic distribution with shape parameter \( p \). The primary difference between the logarithmic and linear scales is that, while the difference in value between linear points of equal distance remains constant (that is, if the space from [latex]0[/latex] to [latex]1[/latex] on the scale is [latex]1[/latex] cm on the page, the distance from [latex]1[/latex] to [latex]2[/latex], [latex]2[/latex] to [latex]3[/latex], etc., will be the same), the difference in value between points on a logarithmic scale will change exponentially. \( X \) has quantile function \( F^{-1} \) given by \frac{\Li_{n+1}(1 - p)}{\ln(p)} = n! Sound . The moments of \( X \) (about 0) are \( R \) is concave upward on \( [0, \infty) \). For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Featured on Meta New Feature: Table Support Assumptions. \[ \E(X^n) = -\frac{1}{\ln(p)} n! The polylogarithm is a power series in \( x \) with radius of convergence is 1 for each \( s \in \R \). The exponential distribution. The curve approaches infinity zero as approaches infinity. Open the random quantile experiment and select the exponential-logarithmic distribution. Let us again consider the graph of the following function: This can be written in exponential form as: Now let us consider the inverse of this function. The top right and bottom left are called semi-log scales because one axis is scaled linearly while the other is scaled using logarithms. In this section, we are only interested in nonnegative integer orders, but the polylogarithm will show up again, for non-integer orders, in the study of the zeta distribution. has the exponential-logarithmic distribution with shape parameter \( p \) and scale parameter \( b \). Vary the shape parameter and note the shape of the distribution and probability density functions. That is, the graph can take on any real number. 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