## 统计代写|统计推断代写Statistical inference代考|Marginal Distributions

The second component of condition [c], relating to the independence of the trials, is defined in terms of a simple relationship between the joint density function $f\left(x_1, x_2, \ldots, x_n ; \boldsymbol{\phi}\right)$ and the density functions of the individual random variables $X_1, X_2, \ldots, X_n$, referred to as the marginal distributions. Let us see how the marginal is related to the joint distribution.

It should come as no surprise to learn that from the joint distribution one can always recover the marginal (univariate) distributions of the individual random variables involved. In terms of the joint cdf, the marginal distribution is derived via a limiting process:
$$F_X(x)=\lim {y \rightarrow \infty} F(x, y) \quad \text { and } \quad F_Y(y)=\lim {x \rightarrow \infty} F(x, y) .$$
Example 4.11 Let us consider the case of the bivariate exponential cdf:
$$F(x, y)=F(x, y)=\left(1-e^{-\alpha x}\right)\left(1-e^{-\beta y}\right), \alpha>0, \beta>0, x>0, y>0 .$$
Given that $\lim {n \rightarrow \infty}\left(e^{-n}\right)=e^{-\infty}=0$, we can deduce that $$F_X(x)=\lim {y \rightarrow \infty} F(x, y)=1-e^{-\alpha x}, x>0, \quad F_Y(y)=\lim {x \rightarrow \infty} F(x, y)=1-e^{-\beta y}, y>0 .$$ Let us see how the marginalization is defined in terms of the density functions. In view of the fact that $$F_X(x)=\lim {y \rightarrow \infty} F(x, y)=\lim {y \rightarrow \infty} \int{-\infty}^x \int_{-\infty}^y f(x, y) d y d x=\int_{-\infty}^x\left\lfloor\int_{-\infty}^{\infty} f(x, y) d y\right\rfloor d x,$$ and the relationship between $F_X(x)$ and $f_x(x)$, we can deduce that
$$f_x(x)=\int_{-\infty}^{\infty} f(x, y) d y, x \in \mathbb{R}X .$$ Similarly, in terms of the joint density function, the marginal density function of $Y$ is derived via $$f_y(y)=\int{-\infty}^{\infty} f(x, y) d x, y \in \mathbb{R}_{Y:}$$
That is, marginalization amounts to integrating out the other random variable.

## 统计代写|统计推断代写Statistical inference代考|Continuous Random Variables

In the case of two continuous random variables $X$ and $Y$, we cannot use the events $A={Y=$ $y}$ and $B={X=x}$ in order to transform (4.20) in terms of density functions, because as we know, in such a case $P(X=x)=0$ and $P(Y=y)=0$ for all $x \in \mathbb{R}, y \in \mathbb{R}$. As in the case of the definition of the joint and marginal density functions, we need to consider events of the form
$$A={X \leq x} \text { and } B={Y \leq y} .$$
However, even in the case of continuous random variables, we would like to be able to refer to the conditional distribution of $Y$ given $X=x$. The way we get around the mathematical difficulties is by way of the conditional cumulative distribution function defined as follows:
$$F_{Y \mid X}(y \mid X=x)=\lim {h \rightarrow 0^{+}} \frac{\mathbb{P}(Y \leq y, x \leq X \leq x+h)}{\mathbb{P}\left(x{-} X_{-} x+h\right)},$$
where $h \rightarrow 0^{+}$reads “as $h$ tends to 0 through values greater than 0 .” After some mathematical manipulations, we can show that
$$F_{Y \mid X}(y \mid X=x)=\lim {h \rightarrow 0^{+}} \frac{\mathbb{P}(Y \leq y, x \leq X \leq x+h)}{\mathbb{P}(x \leq X \leq x+h)}=\int{-\infty}^y \frac{f(x, u)}{f_x(x)} d u .$$
This suggests that in the case of two continuous random variables $X$ and $Y$, we could indeed define the conditional density function as in (4.21) but we should not interpret $f(y \mid x)$ as assigning probabilities because
$$f(. \mid x): \mathbb{R}_Y \rightarrow[0, \infty)$$
As we can see, the conditional density is a proper density function, in so far as, in the case of continuous random variables, it satisfies the properties in Table $4.5$.

## 统计代写|统计推断代写统计推断代考|边际分布

$$F_X(x)=\lim {y \rightarrow \infty} F(x, y) \quad \text { and } \quad F_Y(y)=\lim {x \rightarrow \infty} F(x, y) .$$

$$F(x, y)=F(x, y)=\left(1-e^{-\alpha x}\right)\left(1-e^{-\beta y}\right), \alpha>0, \beta>0, x>0, y>0 .$$

$$f_x(x)=\int_{-\infty}^{\infty} f(x, y) d y, x \in \mathbb{R}X .$$同样，联合密度函数中$Y$的边际密度函数由$$f_y(y)=\int{-\infty}^{\infty} f(x, y) d x, y \in \mathbb{R}_{Y:}$$

## 统计代写|统计推断代写统计推断代考|连续随机变量

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