probabilityandprobabilitydistributions

probabilityandprobabilitydistributions内容摘要：

xample crop yield, maximum temperature.  A discrete variable has a countable set of values. They may be  counts, such as numbers of cyclones  categories, such as much above average, above average, near average, below average, much below average  binary variables, such as rain/no rain Probability distributions  If we measure a random variable many times, we can build up a distribution of the values it can take.  Imagine an underlying distribution of values which we would get if it was possible to take more and more measurements under the same conditions.  This gives the probability distribution for the variable. Discrete probability distributions  A discrete probability distribution associates a probability with each value of a discrete random variable.  Example 1. Random variable has two values Rain/No Rain. P(Rain) = , P(No Rain) = gives a probability distribution.  Example 2. Let X = Number of wet days in a 10 day period. P(X=0) = , P(X=1) = , P(X=2) = , … P(X= 6) = , ... (see Slide 24 for more on this example).  Note that P(rain) + P(No Rain) = 1。 P(X=0) + P(X=1) + P(X=2) + … +P(X= 6) + … P(X= 10) = 1. Continuous probability distributions  Because continuous random variables can take all values in a range, it is not possible to assign probabilities to individual values.  Instead we have a continuous curve, called a probability density function, which allows us to calculate the probability a value within any interval.  This probability is calculated as the area under the curve between the values of interest. The total area under the curve must equal 1. Example: probability distribution for maximum temperature  The next Slide shows an idealized probability density for maximum daily temperature at a station in a particular month.  The total area under the curve is 1.  The area under the curve to the left of 20 is the probability of a max. temperature less than 20176。 C.  The area between 25 and 30 the probability of a max. temp. between 25176。 C and 30176。 C.  The area to right of 32 is the prob. of the max. temp. exceeding 32176。 C. Example: theoretical probability density for maximum temperature 2 0 3 0 4 00 . 00 . 10 . 20 . 30 . 4tf(t)Families of probability distributions  The number of different probability distributions is unlimited. However, certain families of distributions give good approximations to the distributions of many random variables.  Important families of discrete distributions include binomial。