# 统计代写|随机过程代写stochastic process代考|RANDOM SELECTION OF POINTS FROM INTERVALS

#### Doug I. Jones

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## 统计代写|随机过程代写stochastic process代考|RANDOM SELECTION OF POINTS FROM INTERVALS

In Section 1.6, we showed that the probability of the occurrence of any particular point in a random selection of points from an interval $(a, b)$ is 0 . This implies immediately that if $[\alpha, \beta] \subseteq(a, b)$, then the events that the point falls in $[\alpha, \beta],(\alpha, \beta),[\alpha, \beta)$, and $(\alpha, \beta]$ are all equiprobable. Now consider the intervals $\left(a, \frac{a+b}{2}\right)$ and $\left(\frac{a+b}{2}, b\right)$; since $\frac{a+b}{2}$ is the midpoint of $(a, b)$, it is reasonable to assume that
$$p_1=p_2,$$
where $p_1$ is the probability that the point belongs to $\left(a, \frac{a+b}{2}\right)$ and $p_2$ is the probability that it belongs to $\left(\frac{a+b}{2}, b\right)$. The events that the random point belongs to $\left(a, \frac{a+b}{2}\right)$ and $\left(\frac{a+b}{2}, b\right)$ are mutually exclusive and
$$\left(a, \frac{a+b}{2}\right) \cup\left[\frac{a+b}{2}, b\right)=(a, b) ;$$
therefore,
$$p_1+p_2=1$$
This relation and (1.5) imply that
$$p_1=p_2=1 / 2$$

## 统计代写|随机过程代写stochastic process代考|WHAT IS SIMULATION?

Solving a scientific or an industrial problem usually involves mathematical analysis and/or simulation. To perform a simulation, we repeat an experiment a large number of times to assess the probability of an event or condition occurring. For example, to estimate the probability of at least one 6 occurring within four rolls of a die, we may do a large number of experiments rolling a die four times and calculate the number of times that at least one 6 is obtained. Similarly, to estimate the fraction of time that, in a certain bank all the tellers are busy, we may measure the lengths of such time intervals over a long period $X$, add them, and then divide by $X$. Clearly, in simulations, the key to reliable answers is to perform the experiment a large number of times or over a long period of time, whichever is applicable. Since manually this is almost impossible, simulations are carried out by computers. Only computers can handle millions of operations in short periods of time.

To simulate a problem that involves random phenomena, generating random numbers from the interval $(0,1)$ is essential. In almost every simulation of a probabilistic model,we will need to select random points from the interval $(0,1)$. For example, to simulate the experiment of tossing a fair coin, we draw a random number from $(0,1)$. If it is in $(0,1 / 2)$, we say that the outcome is heads, and if it is in $[1 / 2,1)$, we say that it is tails. Similarly, in the simulation of die tossing, the outcomes $1,2,3,4,5$, and 6 , respectively, correspond to the events that the random point from $(0,1)$ is in $(0,1 / 6),[1 / 6,1 / 3),[1 / 3,1 / 2),[1 / 2,2 / 3),[2 / 3,5 / 6)$, and $[5 / 6,1)$.

As discussed in Section 1.7, choosing a random number from a given interval is, in practice, impossible. In real-world problems, to perform simulation we use pseudorandom numbers instead. To generate $n$ pseudorandom numbers from a uniform distribution on an interval $(a, b)$, we take an initial value $x_0 \in(a, b)$, called the seed, and construct a function $\psi$ so that the sequence $\left{x_1, x_2, \ldots, x_n\right} \subset(a, b)$ obtained recursively from
$$x_{i+1}=\psi\left(x_i\right), \quad 0 \leq i \leq n-1,$$
satisfies certain statistical tests for randomness. (Choosing the tests and constructing the function $\psi$ are complicated matters beyond the scope of this book.) The function $\psi$ takes a seed and generates a sequence of pseudorandom numbers in the interval $(a, b)$. Clearly, in any pseudorandom number generating process, the numbers generated are rounded to a certain number of decimal places. Therefore, $\psi$ can only generate a finite number of pseudorandom numbers, which implies that, eventually, some $x_j$ will be generated a second time. From that point on, by (1.7), a pitfall is that the same sequence of numbers that appeared after $x_j$ ‘s first appearance will reappear. Beyond that point, numbers are not effectively random. One important aspect of the construction of $\psi$ is that the second appearance of any of the $x_j$ ‘s is postponed as long as possible.

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|RANDOM SELECTION OF POINTS FROM INTERVALS

$$p_1=p_2,$$

$$\left(a, \frac{a+b}{2}\right) \cup\left[\frac{a+b}{2}, b\right)=(a, b) ;$$

$$p_1+p_2=1$$

$$p_1=p_2=1 / 2$$

## 统计代写|随机过程代写stochastic process代考|WHAT IS SIMULATION?

$$x_{i+1}=\psi\left(x_i\right), \quad 0 \leq i \leq n-1,$$

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