How to calculate the variance of a sample mean distribution with alternating weights?

[JanB]

Hello everyone,

this statement was given in my econometricy textbook. I do understand why the variance of Y bar would be var(Y)/n. Unfortunately, I struggle to understand how one can derive the factor 1.25 from a weighted mean distribution. Could someone please help me out on that one?

 

[blamocur]

Hello everyone,

this statement was given in my econometricy textbook. I do understand why the variance of Y bar would be var(Y)/n. Unfortunately, I struggle to understand how one can derive the factor 1.25 from a weighted mean distribution. Could someone please help me out on that one?
View attachment 37499

Do you know what is the variance of a sum of independent variables?
Then, what are variances of [imath]\frac{1}{2}Y_1[/imath] and [imath]\frac{3}{2} Y_2[/imath] ?