Uniform distribution continuous distributions, random variables

The simplest continuous distribution is the uniform distribution that assigns a constant density function over a region of values from a to b, and assigns zero probability to all other values of the random variable Figure 1.2. 

A bounded continuous random variable with uniform distribution has the probability function... 

To illustrate, consider a uniform cash flow of 1000 per year beginning at some uncertain time m and continuing for a duration of t years. The delay to initiation is uniformly distributed between 6 months and 1 year. The project duration is gamma distributed with mean of 3 years and standard deviation of 1 year the parameters of the gamma distribution yielding these statistics are a = 3 and b = 9. The nominal interest rate is 10% compounded continuously. It is assumed that the initiation time and project dmation are independent random variables. Our problem is to determine the equivalent present value of these cash flows. (This problem is taken from Park and Sharp-Bette (1990, p. 411)... 

As shown in Figure 2.1, the daily average temperature in May appears to be uniformly distributed around a central point situated at about 13 °C, which happens to equal its mean temperature. This bell-shaped curve is common to all processes in which the variability (variance) in the data is random follows a normal, or Gaussian distribution. The continuous mathematical function that characterizes this is of an exponential form ... 

The generation on the computer of a continuous random variable X with cumulative distribution function Fx x) is based on transforming a uniform random variable Y in the interval [0, 1], which can be readily generated. The transformation Y=Fx(X) has an inverse because of the monotone nondecreasing nature of the function Fx, so that X = F Y) further, for any number x, X xoY y = Fx x), so that Fr X x = Pr y < y) = y. Hence Y is uniform. The case of discrete X is left for the student. The generation of random vectors with statistically independent components immediately follows from the preceding discussion. However, the case of correlated components clearly requires further considerations. 

Continuous Uniform Distribution n The form of the uniform distribution over a continuous random variable. 

Uniform Distribution n A probability distribution where the probability in the case of a discrete random variable or the density function in the case of a continuous random variable, X, with values, x, are constant (equal) over an interval, a,b), where x is greater than or equal to a and X less than or equal to b and x is zero outside the interval. The uniform distribution is sometimes referred to as the rectangular distribution since, a plot of its probability or density function resembles a rectangle. When the random variable, X, is discrete, the uniform distribution is referred to as the discrete uniform distribution and has the probability, P(x. ), with the form of ... 

From the cumulative probability distribution, we can generate values of W or x at random according to the specified probability distribution. For example, in the case of a continuous variable, we generate a uniformly distributed number 0 < < 1 using rand. The corresponding random value ofx, generated according to the probability distribution p x) satisfies... 

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