### 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|CONTINUOUS PROBABILITY DISTRIBUTIONS

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• Statistical Inference 统计推断
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Probability Distribution Function, Probability

If the random variable can take on any possible value within the range of outcomes, then the probability distribution is said to be a contimuous random variable. ${ }^{7}$ When a random variable is either the price of or the return on a financial asset or an interest rate, the random variable is assumed to be continuous. This means that it is possible to obtain, for example, a price of $95.43231$ or $109.34872$ and any value in between. In practice, we know that financial assets are not quoted in such a way. Nevertheless, there is no loss in describing the random variable as continuous and in many times treating the return as a continuous random variable means substantial gain in mathematical tractability and convenience. For a continuous random variable, the calculation of probabilities is substantially different from the discrete case. The reason is that if we want to derive the probability that the realization of the random variable lays within some range (i.e., over a subset or subinterval of the sample space), then we cannot proceed in a similar way as in the discrete case: The number of values in an interval is so large, that we cannot just add the probabilities of the single outcomes. The new concept needed is explained in the next section.

A probability distribution function $P$ assigns a probability $P(A)$ for every event $A$, that is, of realizing a value for the random value in any specified subset $A$ of the sample space. For example, a probability distribution function can assign a probability of realizing a monthly return that is negative or the probability of realizing a monthly return that is greater than $0.5 \%$ or the probability of realizing a monthly return that is between $0.4 \%$ and $1.0 \%$

To compute the probability, a mathematical function is needed to represent the probability distribution function. There are several possibilities of representing a probability distribution by means of a mathematical function. In the case of a continuous probability distribution, the most popular way is to provide the so-called probability density function or simply density function.

In general, we denote the density function for the random variable $X$ as $f_{X}(x)$. Note that the letter $x$ is used for the function argument and the index denotes that the density function corresponds to the random variable $X$. The letter $x$ is the convention adopted to denote a particular value for the random variable. The density function of a probability distribution is always nonnegative and as its name indicates: Large values for $f_{X}(x)$ of the density function at some point $x$ imply a relatively high probability of realizing a value in the neighborhood of $x$, whereas $f_{X}(x)=0$ for all $x$ in some interval $(a, b)$ implies that the probability for observing a realization in $(a, b)$ is zero.

Figure $1.1$ aids in understanding a continuous probability distribution. The shaded area is the probability of realizing a return less than $b$ and greater than $a$. As probabilities are represented by areas under the density function, it follows that the probability for every single outcome of a continuous random variable always equals zero. While the shaded area

in Figure $1.1$ represents the probability associated with realizing a return within the specified range, how does one compute the probability? This is where the tools of calculus are applied. Calculus involves differentiation and integration of a mathematical function. The latter tool is called integral calculus and involves computing the area under a curve. Thus the probability that a realization from a random variable is between two real numbers $a$ and $b$ is calculated according to the formula,
$$P(a \leq X \leq b)=\int_{a}^{b} f_{X}(x) d x$$
The mathematical function that provides the cumulative probability of a probability distribution, that is, the function that assigns to every real value $x$ the probability of getting an outcome less than or equal to $x$, is called the cumulative distribution function or cumulative probability function or simply distribution function and is denoted mathematically by $F_{X}(x)$. A cumulative distribution function is always nonnegative, nondecreasing, and as it represents probabilities it takes only values between zero and one. ${ }^{8} \mathrm{An}$ example of a distribution function is given in Figure 1.2.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|The Normal Distribution

The class of normal distributions, or Gaussian distributions, is certainly one of the most important probability distributions in statistics and due to some of its appealing properties also the class which is used in most applications in finance. Here we introduce some of its basic properties.

The random variable $X$ is said to be normally distributed with parameters $\mu$ and $\sigma$, abbreviated by $X \in N\left(\mu, \sigma^{2}\right)$, if the density of the random

$$f_{X}(x)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}}, x \in \mathbb{R} \text {. }$$
The parameter $\mu$ is called a location parameter because the middle of the distribution equals $\mu$ and $\sigma$ is called a shape parameter or a scale parameter. If $\mu=0$ and $\sigma=1$, then $X$ is said to have a standard normal distribution.

An important property of the normal distribution is the location-scale invariance of the normal distribution. What does this mean? Imagine you have random variable $X$, which is normally distributed with the parameters $\mu$ and $\sigma$. Now we consider the random variable $Y$, which is obtained as $Y=$ $a X+b .$ In general, the distribution of $Y$ might substantially differ from the distribution of $X$ but in the case where $X$ is normally distributed, the random variable $Y$ is again normally distributed with parameters and $\bar{\mu}=a \mu+b$ and $\bar{\sigma}=a \sigma$. Thus we do not leave the class of normal distributions if we multiply the random variable by a factor or shift the random variable. This fact can be used if we change the scale where a random variable is measured: Imagine that $X$ measures the temperature at the top of the Empire State Building on January 1, 2008, at 6 A.M. in degrees Celsius. Then $Y=\frac{9}{5} X+32$ will give the temperature in degrees Fahrenheit, and if $X$ is normally distributed, then $Y$ will be too.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Exponential Distribution

The exponential distribution is popular, for example, in queuing theory when we want to model the time we have to wait until a certain event takes place. Examples include the time until the next client enters the store, the time until a certain company defaults or the time until some machine has a defect.

As it is used to model waiting times, the exponential distribution is concentrated on the positive real numbers and the density function $f$ and the cumulative distribution function $F$ of an exponentially distributed random variable $\tau$ possess the following form:
$$f_{\mathrm{r}}(x)=\frac{1}{\beta} e^{-\frac{x}{\beta}}, x>0$$
and
$$F_{\mathrm{r}}(x)=1-e^{-\frac{x}{\beta}}, x>0 .$$

In credit risk modeling, the parameter $\lambda=1 / \beta$ has a natural interpretation as hazard rate or default intensity. Let $\tau$ denote an exponential distributed random variable, for example, the random time (counted in days and started on January 1, 2008) we have to wait until Ford Motor Company defaults. Now, consider the following expression:
$$\lambda(\Delta t)=\frac{P(\tau \in(t, t+\Delta t] \mid \tau>t)}{\Delta t}=\frac{P(\tau \in(t, t+\Delta t])}{\Delta t P(\tau>t)} .$$
where $\Delta t$ denotes a small period of time.
What is the interpretation of this expression? $\lambda(\Delta t)$ represents a ratio of a probability and the quantity $\Delta t$. The probability in the numerator represents the probability that default occurs in the time interval $(t, t+\Delta t]$ conditional upon the fact that Ford Motor Company survives until time $t$. The notion of conditional probability is explained in section 1.6.1.

Now the ratio of this probability and the length of the considered time interval can be denoted as a default rate or default intensity. In applications different from credit risk we also use the expressions hazard or failure rate.
Now, letting $\Delta t$ tend to zero we finally obtain after some calculus the desired relation $\lambda=1 / \beta$. What we can see is that in the case of an exponentially distributed time of default, we are faced with a constant rate of default that is independent of the current point in time $t$.

Another interesting fact linked to the exponential distribution is the following connection with the Poisson distribution described earlier. Consider a sequence of independent and identical exponentially distributed random variables $\tau_{1}, \tau_{2}, \ldots$ We can think of $\tau_{1}$, for example, as the time we have to wait until a firm in a high-yield bond portfolio defaults. $\tau_{2}$ will then represent the time between the first and the second default and so on. These waiting times are sometimes called interarrival times. Now, let $N_{t}$ denote the number of defaults which have occurred until time $t \geq 0$. One important probabilistic result states that the random variable $N_{t}$ is Poisson distributed with parameter $\lambda=t / \beta$.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Exponential Distribution

Fr(X)=1b和−Xb,X>0

Fr(X)=1−和−Xb,X>0.

λ(Δ吨)=磷(τ∈(吨,吨+Δ吨]∣τ>吨)Δ吨=磷(τ∈(吨,吨+Δ吨])Δ吨磷(τ>吨).

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