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Mean as center of Gravity. Let X be a discrete random variable with a finite number of number of possible values say $x_1,x_2.....x_m$For convenience, set $P_K=P(X=x_k),$for K = 1, 2.....m . Think of a horizontal axis as a seesaw and each $P_K$as a mass placed at point $x_k$on the seesaw. The center of gravity of those masses is defined to be the point c on the horizontal axis at which a fulcrum could be placed to balance the seesaw.

Relative to the center of gravity , The torque acting on the seesaw by the mass $P_K$is proportional to the product of that mass with the signed distance of the point $X_k$From c That is , to ($(x_k-c)$. $P_k$show that the center of gravity equal the mean of the random variable X ( hint: To balance, the total torque acting on the seesaw must be 0)

Step by step solution

01

Step 1. Given information. 

Set $P_K=P(X=x)$for $k=1,2,3.....m$and consider the possible values of a random variable X to be $x_1,x_2......x_m$. Consider the seesaw below, with each mass $P_K$placed at point $X_K$and the center of gravity at point c. $(x_k-c)p_k$is the torque exerted on the seesaw by mass $p_k$.

A balanced seesaw has a total torque of zero.

02

Step 2. The center of gravity is equal to the mean of the random variable X.

An event's total probability is:

$\underset{}{鈭�}P(X=x)=1$

A random variable's expected value is given by:

$渭=\underset{}{鈭�}xP(X=x)$

The seesaw is balanced if the total torque acting on it equals:

$$\begin{gathered} \underset{k鈥�1}{\overset{m}{鈭�}}(x_k-c)p_k=0 \\ \underset{k鈥�1}{\overset{m}{鈭�}}(x_k路p_k-c路p_k)=0 \\ \underset{k鈥�1}{\overset{m}{鈭�}}(x_k路p_k-\underset{k鈥�1}{\overset{m}{鈭�}}c路p_k)=0 \\ {渭}_x-\underset{k鈥�1}{\overset{m}{鈭�}}c路p_k=0 \\ {渭}_x-c\underset{k鈥�1}{\overset{m}{鈭�}}路P(X=x_k)=0 \\ {渭}_x-c路1=0 \\ {渭}_x-c路1=0 \\ {渭}_x-c=0 \\ {渭}_x=c \end{gathered}$$

As a result, the random variable's mean equals the centre of gravity.

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