The Binomial Distribution二项分布.docVIP

The Uniform and Normal Distribution When a random variable X can take on any value in an interval, we call it a continuous random variable and describe its outcomes using a probability density function. Instead of talking about the probability that X takes on a specific value P(X=x) as we do in the discrete case, we talk about the probability that X falls in an interval such as [c, d] and write it as . The probability that X takes on a specific value in the continuous case is 0. The probability that X falls in an interval [c, d] is determined by calculating the area under the density function that is bounded by c, d, and the x-axis. In the continuous case, probabilities correspond to areas. Uniform Distribution The uniform distribution is possibly the simplest example of a continuous random variable. Its probability density function (“density” for short) is given by The graph of this density is shown below: Uniform Density Notice that the height of the elevated horizontal line is precisely so that the total area under the curve is 1. The mean of a uniform random variable is . The variance is . Example. Suppose U follows a uniform distribution on [3,8] (usually written U[3,8]). (a) What is the probability that U falls between 4 and 6? (b) What is the probability that U falls between 0 and 5? (a) The Normal Distribution The Normal distribution is the most important continuous distribution in all of statistics. The “bell curve,” as it is frequently referred to, has a density function given by for , where and are parameters () which affect the shape of the distribution. The choice of Greek letters is prophetic. If X is a random variable whose density function is , then X has a normal distribution with mean and variance (equivalently, the standard deviation of X is ). Notation: A normal random variable X with mean and variance is described using the shorthand notation . The tilde (~) is read “is distributed as.” Calculations with the Nor