Avg. of sum of independent variables

In summary, the conversation discusses the probability of multiple variables and the assumption that dP(Y) = p(Y) dY is not used. The correct expression for P(Y) should include integration of all the variables involved. This leads to the conclusion that the assumption is compatible with the original probability distribution.
  • #1
Pushoam
962
52

Homework Statement


upload_2017-9-11_10-15-19.png


Homework Equations

The Attempt at a Solution


The probability that ##X_1 ## is between ## X_1 ## and ## X_1 + dX_1 ## and ##X_2 ## is between ## X_2 ## and ## X_2 + dX_2 ## and so on till the nth variable is
dP(##X_1,
X_2, ..., X_n) = p ( X_1) p( x_2) p(X_3)...p(X_n) dX_1 dX_2 ...dX_n
\\ dP(Y) = p(Y) dY
\\<Y>= \int Yp(Y) dY
\\ assuming dP(y) =
dP(X_1,
X_2, ..., X_n)

\\= \int ( X_1 +X_2 + ... + X_n) p ( X_1) p( x_2) p(X_3)...p(X_n) dX_1 dX_2 ...dX_n
\\ = \int X_1 p ( X_1)dX_1+ \int X_2 p ( X_2)dX_2+ ...+\int X_n p ( X_n)dX_n
\\= <X_1> + <X_2>+...+<X_n>
\\= n <X>##
Is the assumption o.k?
 
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  • #2
You are never really using that assumption, you are using
$$
\langle Y \rangle = \int Y(x_1,\ldots,x_n) P(x_1,\ldots, x_n) dx_1 \ldots dx_n.
$$
If you want to express ##P(Y)## as a one-variable function you need to integrate out the ##X_i## variables, essentially
$$
P(Y) = \int \delta(Y-x_1-\ldots-x_n) P(x_1,\ldots,x_n) dx_1 \ldots dx_n.
$$
Note that this leads to
$$
\langle f(Y)\rangle = \int f(y) P(y) dy = \int f(y) \delta(y-x_1-\ldots-x_n) P(x_1,\ldots,x_n) dx_1 \ldots dx_n dy
= \int f(x_1+\ldots+x_n) P(x_1,\ldots,x_n) dx_1 \ldots dx_n,
$$
so it is compatible with the original probability distribution.
 
  • #3
Thank you for this insight.
 

Related to Avg. of sum of independent variables

1. What does "Avg. of sum of independent variables" mean?

"Avg. of sum of independent variables" refers to the average value obtained by adding together a group of independent variables and then dividing that sum by the number of variables. This calculation is commonly used in statistics to measure the overall average or central tendency of a set of data points.

2. How is the "Avg. of sum of independent variables" calculated?

The "Avg. of sum of independent variables" is calculated by first adding together all the values of the independent variables in a set of data and then dividing that sum by the total number of variables. This calculation is typically represented by the formula (x1 + x2 + ... + xn) / n, where x represents the individual values and n represents the number of variables.

3. What is the importance of calculating the "Avg. of sum of independent variables"?

The "Avg. of sum of independent variables" is important because it provides a measure of the central tendency or average value of a set of data points. This can help researchers and scientists to better understand and analyze the data, as well as make predictions and draw conclusions based on the average value.

4. Can the "Avg. of sum of independent variables" be used for any type of data?

Yes, the "Avg. of sum of independent variables" can be used for any type of data as long as the variables are independent of each other. This means that the values of one variable do not depend on or affect the values of another variable. If the variables are not independent, then a different calculation, such as the average of dependent variables, may need to be used.

5. How is the "Avg. of sum of independent variables" different from the "Avg. of sum of dependent variables"?

The "Avg. of sum of independent variables" and the "Avg. of sum of dependent variables" are different in that the former calculates the average value of a set of independent variables, while the latter calculates the average value of a set of dependent variables. Dependent variables are those that are affected by or depend on the values of other variables in the data set, while independent variables are not affected by other variables. The calculation for the "Avg. of sum of dependent variables" is typically represented by the formula (x1 * y1 + x2 * y2 + ... + xn * yn) / n, where x and y represent the values of two dependent variables and n represents the number of variables.

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