[Statistics review] Four arithmetic operations of random variables

Introduction

Now that I've started reviewing ** Statistics **, I'll summarize my own interpretation as a memorandum. This time, I will summarize the most basic ** four arithmetic operations of random variables ** in statistics, and in the future I will summarize ** estimation, testing, multivariate analysis, and Bayesian statistics **. I will try to be helpful for those who want to review statistics and those who want to check their understanding of statistics.

What is Statistical Inference?

What is inference statistics in the first place? A famous anecdote about inference statistics is the story of ** Poincare and Bakery **.

The anecdote is as follows. Poincare is a familiar bakery, and he often bought bread weighing 1000g. However, Poincare, who felt that the weight was being cheated, decided to weigh the bread he bought every time. A year later, Poincare confirmed that the bread he had bought so far had a normal distribution with an average weight of 950g, and he found out the fraud.

Now, how much bread did Poincaré actually need to buy to detect the injustice of the bakery? Using statistical interval estimation, we can see that ** buying about 70 pieces of bread is 99.98% certainty that the bakery is cheating ** (quite cautious). * Strictly speaking, it is necessary to determine the number of specimens in advance in order to avoid making the sampling intentional.

Four arithmetic operations in statistics

Before we go to interval estimation of statistics, we need to understand ** formulas of four arithmetic operations in statistics **. We are deriving the formula, but if it is troublesome, you can understand it to some extent by checking only the results and comments.

For convenience, define the following symbols and formulas: Also, for the sake of simplicity, we will assume that the weight of the bread is a discrete value that takes only an integer value, but in the case of a continuous value (real number), just set $ \ sum_ {} $ to $ \ int $.

** Definition of symbols and formulas ** </ font> ・ $ X = (x_1, x_2, x_3 ・ ・ ・ x_n): $ ** Random variable ** (values that can take the weight of bread, for example 900g, 901g, 902g ・ ・ ・ 1000g, 1001g, 1002g, etc.)

\begin{align}
E(X+Y)&=\sum_{j,k}(x_j+y_k)f(x_j,y_k)\\
&=\sum_{j,k}x_jf(x_j,y_k)+\sum_{j,k}y_kf(x_j,y_k)\\
&=\sum_{j}x_jf(x_j)\sum_{k}f(y_k)+\sum_{k}y_kf(y_k)\sum_{j}f(x_j)\\
&=\sum_{j}x_jf(x_j)+\sum_{k}y_kf(y_k)\\
&=E(X)+E(Y)
\end{align}

\begin{align}
E(X×Y)=&\sum_{j,k}\bigl(x_jy_kf(x_j,y_k)\bigr)\\

=&\sum_{j}x_jf(x_j)\sum_{k}y_kf(y_k)\\
=&E(X)E(Y)
\end{align}

\begin{align}
E(a×X)&=\sum_{j}\bigl(a(x_j)f(x_j)\bigr)\\
&=ax_1f(x_1)+ax_2f(x_2)・ ・ ・\\
&=aE(X)
\end{align}

\begin{align}
V(X+Y)=&\sum_{j,k}\bigl(x_j+y_k-(μ_X+μ_Y)\bigr)^2f(x_j,y_k)\\
=&\sum_{j,k}\bigl(x_j^2+y_k^2+μ_X^2+μ_Y^2+2x_jy_k-2x_jμ_X-2x_jμ_Y-2y_kμ_X-2y_kμ_Y+2μ_Xμ_Y\bigr)f(x_j,y_k)\\
=&\sum_{j,k}\bigl(x_j-μ_X\bigr)^2f(x_j,y_k)+\sum_{j,k}\bigl(y_k-μ_Y\bigr)^2f(x_j,y_k)+2\sum_{j,k}\bigl(x_j-μ_X\bigr)\bigl(y_k-μ_Y\bigr)f(x_j,y_k)\\
=&\sum_{j}\bigl(x_j-μ_X\bigr)^2f(x_j)+\sum_{k}\bigl(y_k-μ_Y\bigr)^2f(y_k)+2\sum_{j}\bigl(x_j-μ_X\bigr)f(x_j)\sum_{k}\bigl(y_k-μ_Y\bigr)f(y_k)\\
=&V(X)+V(Y)
\end{align}

\begin{align}
V(X×Y)=&\sum_{j,k}\bigl(x_jy_k-μ_Xμ_Y\bigr)^2f(x_j,y_k)\\
=&\sum_{j,k}\bigl((x_jy_k)^2-2x_jy_kμ_Xμ_Y+(μ_Xμ_Y)^2\bigr)f(x_j,y_k)\\
=&\sum_{j,k}(x_jy_k)^2f(x_j,y_k)-μ_Xμ_Y\sum_{j,k}2x_jy_kf(x_j,y_k)+(μ_Xμ_Y)^2\\
=&\sum_{j,k}(x_jy_k)^2f(x_j,y_k)-2(μ_Xμ_Y)^2+(μ_Xμ_Y)^2\\
=&\sum_{j}x_j^2f(x_j)\sum_{k}y_k^2f(y_k)-(μ_Xμ_Y)^2 \\
=&E(X^2)E(Y^2)-(μ_Xμ_Y)^2\\
=&\bigl(V(X)+μ_X^2\bigr)\bigl(V(Y)+μ_Y^2\bigr)-(μ_Xμ_Y)^2　※V(X)=\sum_{k}(x_k-μ_X)^2f(x_k)=E(X^2)-μ_X^2\\
=&V(X)V(Y)+μ_Y^2V(X)+μ_X^2V(Y)
\end{align}

\begin{align}
V(a×X)=&\sum_{j}(ax_j-μ_{ax})^2f(x_j)\\
=&(ax_1-aμ_x)^2+(ax_2-aμ_x)^2+(ax_3-aμ_x)^2+・ ・ ・\\
=&a^2\sum_{j}(x_j-μ_{x})^2f(x_j)\\
=&a^2V(X)
\end{align}