MHB Statistics - Finding a relationship?

AI Thread Summary
The discussion focuses on establishing relationships between the means and standard deviations of two variables, y and x, defined by the equation y_j = ax_j + b. It is clarified that the mean of y, denoted as \(\overline{y}\), can be expressed as \(\overline{y} = a\overline{x} + b\). Additionally, the standard deviation of y, \(\sigma_y\), is found to be \(\sigma_y = a\sigma_x\). This indicates that the mean of y is a linear transformation of the mean of x, while the standard deviation of y is scaled by the constant a. Overall, the relationships derived align with statistical intuition regarding linear transformations.
shamieh
Messages
538
Reaction score
0
let a and b be constants and let $$y_j = ax_j+b$$ for $$j = 1,2...n$$. What are the relationships between the means of y and x, and the standard deviations of y and x?

I'm not sure what they are wanting here?
 
Mathematics news on Phys.org
Let's look at the means first...we know:

$$\overline{y}=\frac{1}{n}\sum_{j=1}^n\left(y_j\right)$$

Can you use the definition of $y_j$ and the properties of sums to obtain a relationship between $\overline{y}$ and $\overline{y}$?

Once you have the above, then look at:

$$\sigma_y=\sqrt{\frac{\sum\limits_{j=1}^n\left(y_j-\overline{y}\right)^2}{n}}$$

Replace $y_j$ and $\overline{y}$ with the expressions in terms of $x_j$ and $\overline{x}$, and you should be able to establish a relationship between $\sigma_y$ and $\sigma_x$. :D
 
would it be that they are both squared?
 
I don't really understand. This is my first statistic class. is ybar supposed to be the symbol for the mean?
 
MarkFL said:
Let's look at the means first...we know:

$$\overline{y}=\frac{1}{n}\sum_{j=1}^n\left(y_j\right)$$

Can you use the definition of $y_j$ and the properties of sums to obtain a relationship between $\overline{y}$ and $\overline{y}$?

Once you have the above, then look at:

$$\sigma_y=\sqrt{\frac{\sum\limits_{j=1}^n\left(y_j-\overline{y}\right)^2}{n}}$$

Replace $y_j$ and $\overline{y}$ with the expressions in terms of $x_j$ and $\overline{x}$, and you should be able to establish a relationship between $\sigma_y$ and $\sigma_x$. :D

Sorry for multiple posts. I think I see it now. the deviation of the $i^{th}$ observation, $y_i$, from the sample mean $\overline{y}$ is the difference between them $y_i - \overline{y}$
 
Yes, the bar over a variable represents the mean.

This is what I was suggesting you do:

$$\overline{y}=\frac{1}{n}\sum_{j=1}^n\left(y_j\right)=\frac{1}{n}\sum_{j=1}^n\left(ax_j+b\right)=a\frac{1}{n}\sum_{j=1}^n\left(x_j\right)+\frac{1}{n}bn=a\overline{x}+b$$

Now for the deviation:

$$\sigma_y=\sqrt{\frac{\sum\limits_{j=1}^n\left(y_j-\overline{y}\right)^2}{n}}=\sqrt{\frac{\sum\limits_{j=1}^n\left(ax_j+b-a\overline{x}-b\right)^2}{n}}=a\sqrt{\frac{\sum\limits_{j=1}^n\left(x_j-\overline{x}\right)^2}{n}}=a\sigma_x$$

Both of these results should agree nicely with intuition too. :D
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top