Analytic determination of Expectation, variance

[Eren10]

Hi,

I want to proof what the distribution will be when I apply a normal distributed x to a linear function y = a*x + b. What will be the mean and the variance of y ?

The expectations can be calculated than with this formula ( probably with this formula what i want can be proofed with substitution):

E_x [y] = integral ( y(x)*rho_x(x)dx) , x is the randomly event , normal distributed, y = a*x+b.

After this I want to proof that for n - dimensional parameter the variance $$\sigma = \sum\sigma^2$$

for example I want also proof that the sum of normal distributed parameters X and Y is also normal distributed.

can someone at least advice which book I need to see how this kind of things are proofed.

 

[mathman]

These are elementary probability calculations. E(y)=aE(x)+b, V(y)=a²V(x)

x+y=(a+1)*x + b, so the above can be used with a+1 replacing a.

 

[SW VandeCarr]

Hi,


After this I want to proof that for n - dimensional parameter the variance $$\sigma = \sum\sigma^2$$

Can you tell me what these means? A standard deviation is not defined this way. Did you mean $$\sigma^{2}$$ for the left side term? If so it needs a subscript to distinguish it from the right side.

 

[Eren10]

These are elementary probability calculations. E(y)=aE(x)+b, V(y)=a²V(x)

x+y=(a+1)*x + b, so the above can be used with a+1 replacing a.

Thank you for your reply. I am reading now google book about elementary probability. For simple cases I have to show how they came to that relation and this will help me to understand the idea behind it better.

Can you tell me what these means? A standard deviation is not defined this way. Did you mean $$\sigma^{2}$$ for the left side term? If so it needs a subscript to distinguish it from the right side.

Thank you for your reply. I made there a mistake. assume a function f(X,Y,X) , X,Y,Z are randomly variables normal distribution. What I will do is: apply the random variables to f separately and then find the variance of f , thus for each variable. and hereafter the variances will be summed. the total variance is:

$$\sigma^{2}_{total} = \Sigma\sigma^{2}_{i}$$ I want to show this by applying the basic principles and than I will apply it at an experiment.

 

[chiro]

Hi,

I want to proof what the distribution will be when I apply a normal distributed x to a linear function y = a*x + b. What will be the mean and the variance of y ?

The expectations can be calculated than with this formula ( probably with this formula what i want can be proofed with substitution):

E_x [y] = integral ( y(x)*rho_x(x)dx) , x is the randomly event , normal distributed, y = a*x+b.

After this I want to proof that for n - dimensional parameter the variance $$\sigma = \sum\sigma^2$$

for example I want also proof that the sum of normal distributed parameters X and Y is also normal distributed.

can someone at least advice which book I need to see how this kind of things are proofed.

I don't know if this is what you want but with moment generating functions you can prove that the sum of two normal distributions is another normal distribution. In fact you can use MGF's to show what a lot of different types of added distributions are (like say Poisson + Poisson or Gamma + Gamma etc).