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

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• Longitudinal Data Analysis 纵向数据分析
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## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Conditional Probability

A useful concept in understanding the relationship between multiple random variables is that of conditional probability. Consider the returns on the stocks of two companies in one and the same industry. The future return $X$ on the stocks of company 1 is not unrelated to the future return $Y$ on the stocks of company 2 because the future development of the two companies is driven to some extent by common factors since they are in one and the same industry. It is a reasonable question to ask, what is the probability that the future return $X$ is smaller than a given percentage, e.g. $X \leq-2 \%$, on condition that $Y$ realizes a huge loss, e.g. $Y \leq-10 \%$ ? Essentially, the conditional probability is calculating the probability of an event provided that another event happens. If we denote the first event by $A$ and the second event by $B$, then the conditional probability of $A$ provided that $B$ happens, denoted by $P(A \mid B)$, is given by the formula,
$$P(A \mid B)=\frac{P(A \cap B)}{P(B)}$$
which is also known as the Bayes formula. According to the formula, we divide the probability that both events $A$ and $B$ occur simultaneously, denoted by $A \cap B$, by the probability of the event $B$. In the two-stock example, the formula is applied in the following way,
$$P(X \leq-2 \% \mid Y \leq-10 \%)=\frac{P(X \leq-2 \%, Y \leq-10 \%)}{P(Y \leq-10 \%)}$$
Thus, in order to compute the conditional probability, we have to be able to calculate the quantity
$$P(X \leq-2 \%, Y \leq-10 \%)$$
which represents the joint probability of the two events.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Definition of Joint Probability Distributions

A portfolio or a trading position consists of a collection of financial assets. Thus, portfolio managers and traders are interested in the return on a portfolio or a trading position. Consequently, in real-world applications, the interest is in the joint probability distribution or joint distribution of more than one random variable. For example, suppose that a portfolio consists of a position in two assets, asset 1 and asset 2 . Then there will be a probability distribution for (1) asset 1 , (2) asset 2, and (3) asset 1 and asset 2. The first two distributions are referred to as the marginal probability distributions or marginal distributions. The distribution for asset 1 and asset 2 is called the joint probability distribution.

Like in the univariate case, there is a mathematical connection between the probability distribution $P$, the cumulative distribution function $F$, and the density function $f$ of a multivariate random variable (also called a random vector) $X=\left(X_{1}, \ldots, X_{n}\right)$. The formula looks similar to the equation we presented in the previous chapter showing the mathematical connection between a probability density function, a probability distribution, and a cumulative distribution function of some random variable $X$ :
\begin{aligned} P\left(X_{1} \leq t_{1}, \ldots, X_{n} \leq t_{n}\right) &=F_{X}\left(t_{1}, \ldots, t_{n}\right) \ &=\int_{-\infty}^{t_{1}} \ldots \int_{-\infty}^{t_{n}} f_{X}\left(x_{1}, \ldots, x_{n}\right) d x_{1} \ldots d x_{n} \end{aligned}
The formula can be interpreted as follows. The joint probability that the first random variable realizes a value less than or equal to $t_{1}$ and the second less than or equal to $t_{2}$ and so on is given by the cumulative distribution function $F$. The value can be obtained by calculating the volume under the density function $f$. Because there are $n$ random variables, we have now $n$ arguments for both functions: the density function and the cumulative distribution function.

It is also possible to express the density function in terms of the distribution function by computing sequentially the first-order partial derivatives of the distribution function with respect to all variables,
$$f_{X}\left(x_{1}, \ldots, x_{n}\right)=\frac{\partial^{n} F_{X}\left(x_{1}, \ldots, x_{n}\right)}{\partial x_{1} \ldots \partial x_{n}}$$

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Dependence of Random Variables

Typically, when considering multivariate distributions, we are faced with inference between the distributions; that is, large values of one random variable imply large values of another random variable or small values of a third random variable. If we are considering, for example, $X_{1}$, the height of a randomly chosen U.S. citizen, and $X_{2}$, the weight of this citizen, then large values of $X_{1}$ tend to result in large values of $X_{2}$. This property is denoted as the dependence of random variables and a powerful concept to measure dependence will be introduced in a later section on copulas.

The inverse case of no dependence is denoted as stochastic independence. More precisely, two random variables are independently distributed if and only if their joint distribution given in terms of the joint cumulative distribution function $F$ or the joint density function $f$ equals the product of their marginal distributions:
\begin{aligned} &F_{X}\left(x_{1}, \ldots, x_{n}\right)=F_{X_{1}}\left(x_{1}\right) \ldots F_{X_{n}}\left(x_{n}\right) \ &f_{X}\left(x_{1}, \ldots, x_{n}\right)=f_{X_{1}}\left(x_{1}\right) \ldots f_{X_{n}}\left(x_{n}\right) \end{aligned}
In the special case of $n=2$, we can say that two random variables are said to be independently distributed, if knowing the value of one random variable does not provide any information about the other random variable. For instance, if we assume in the example developed in section 1.6.1 that the two events $X \leq-2 \%$ and $Y \leq-10 \%$ are independent, then the conditional probability in equation (1.1) equals
\begin{aligned} P(X \leq-2 \% \mid Y \leq-10 \%) &=\frac{P(X \leq-2 \%) P(Y \leq-10 \%)}{P(Y \leq-10 \%)} \ &=P(X \leq-2 \%) \end{aligned}
Indeed, under the assumption of independence, the event $Y \leq-10 \%$ has no influence on the probability of the other event.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Definition of Joint Probability Distributions

FX(X1,…,Xn)=∂nFX(X1,…,Xn)∂X1…∂Xn

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Dependence of Random Variables

FX(X1,…,Xn)=FX1(X1)…FXn(Xn) FX(X1,…,Xn)=FX1(X1)…FXn(Xn)

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