MTH-102复习3

连续随机变量部分

Lec7

1. Continuous random variables

   1. Probability density function (pdf)

      Let be a random variable over the outcome space which is an interval or union of intervals. is called a continuous random variable if there exists an integrable function satisfying the following:
      (a)
      (b)
      (c)
      is called the Probability density function (pdf) of

   2. Character

      对于一个连续随机变量X，它在实数域上任意一特定点的概率都是0
      因此

   3. Cumulative distribution function (cdf)

      The Cumulative distribution function (cdf) of a continuous random variable with the pdf is given by Although , we write for consistency with the cdf of discrete random variables.

      • Properties

        • is continuous, and for values for which the derivative exists,
        • For any

2. Mean and variance

   1. Mean of the continuous random variable

      Remark. max not exist
      

      2. Variance

      • Properties

        • The variance is always nonnegative
        • The square root of , i.e. , is called the standard deviation of
        • Let be a random variable, , and be constants. Consider as a linear function of . Then

3. Uniform distribution

   A random variable is said to be uniformly distributed over the interval if its probability density function is given byThe cdf of is The mean and variance of are

Lec8

1. Exponential distribution

   • Definition

     If a continuous random variable is said to be an exponential random variable. The pdf is
     The cdf of is
     The mean and variance are

   • Applications

     指数分布通常应用于预测直到某个特定事件发生前的时间长度。
     泊松分布通常应用于预测在某个特定的时间间隔中事件发生的次数。
     事实上，泊松分布事件发生的间隔时间是服从指数分布的，如果和是泊松分布的参数和相关指数分布的参数，那么

   • Memoryless property

     指数分布是无记忆的，即当
     证明：因为是指数分布，所以因此如此可得
     当 是等待一辆公交车的时间，并且是遵循指数分布，拥有参数的连续随机变量。
     乘客A比乘客B更早到达公交车站，当乘客B到达车站时，乘客A已经等待了s分钟，并且车还没来。
     已知乘客A已经等待了s分钟，那么他再等t分钟的概率是，乘客B再等t分钟的概率是因为指数分布的无记忆，他们会再等待t分钟的概率相等，同时这也符合现实规律。

2. Normal distribution

   • Definition

     当一个随机变量符合正态分布，其pdf是
     当是阐述，我们可以说
     那么
     令我们可以得到
     X的期望和方差分别是

   • Standard normal distribution

     当我们称Z遵循标准正态分布，这时，Z的cdf时
     
     因为当任意实数z存在时
     由此可得:

   • Normal distribution theorem

     1. 当X遵循正态分布那么
     2. 当X遵循正态分布那么对于任何
     3. 当对于任何

3. Rayleigh distribution

   • Definition

     瑞丽分布的动机是测量射击距离与射点到靶心的距离。当是射点的坐标，并且服从正态分布拥有参数为的瑞丽分布

   • Rayleigh distribution

     • The pdf
     • The cdf
     • The mean and variance