Yes. the conditional expectation of X given Y is simply the expectation of X if X and Y are uncorrelated. This is a consequence of one of the properties of conditional expectation.

the statistically independent random variables are uncorrelated but the converse is not true ,i want a counter example,

If I have two source of noise let as say two laser diodes so the pink noise that generate fro both of them is it correlated or uncorrelated

An uncorrelated data sequence refers to a series of data points where there is no statistical relationship or dependency between them. In other words, the value of one data point does not provide any information about the value of another, indicating that their correlation coefficient is close to zero. This property is often important in statistical analysis and signal processing, as it suggests that the data points can be treated independently. Uncorrelated sequences are commonly used in various applications, such as random number generation and noise analysis.

to convert raw data of correlated variables to data matrix of uncorrelated variables (Principal Component)

Yes it is. That is actually true for all random vars, assuming the covariance of the two random vars is zero (they are uncorrelated).

Correlation is a way of testing whether there is a linear relationship between two variables of their transformed versions. If there is no correlation then all that it means is that there is no linear relationship between the two or between their transformed versions. It certainly does not mean that there is no relationship. For example, if y = a*x^2 + b where a and b are constants then, because of symmetry, the correlation between x and y is 0.

In statistics, the Gauss-Markov theorem states that in a linear regression model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator (BLUE) of the coefficients is given by the ordinary least squares (OLS) estimator, provided it exists.

In mathematics, "orthogonal" means perpendicular or independent. In linear algebra, vectors are orthogonal if their dot product is zero, indicating they are at right angles to each other. In statistics, orthogonal variables are uncorrelated, making them useful for multi-variable analysis.

PCAs, or Principal Component Analysis, is a statistical technique used for dimensionality reduction and data analysis. It transforms a dataset into a set of orthogonal (uncorrelated) variables called principal components, which capture the most variance in the data. This method helps simplify complex datasets while retaining essential information, making it valuable in fields like machine learning, image processing, and finance for visualization and feature extraction.

This form of validity is known as discriminant validity. It assesses whether a measure is indeed distinct from other measures that assess different constructs, indicating that the test is not measuring something else inadvertently. Discriminant validity is often evaluated through statistical analyses, such as correlation coefficients, demonstrating that the measure does not correlate highly with unrelated constructs.

Two-stage least squares (2SLS) is used instead of ordinary least squares (OLS) when there is concern about endogeneity in the regression model, such as when an independent variable is correlated with the error term. This typically arises in the presence of omitted variable bias, measurement error, or simultaneous causality. 2SLS helps to provide consistent estimators by using instrumental variables that are correlated with the endogenous explanatory variables but uncorrelated with the error term. In contrast, OLS is appropriate when all variables are exogenous and there are no such concerns.

Correlated noise refers to a type of noise in which the fluctuations at one point in time or space are related to fluctuations at another, creating a dependency between them. Unlike white noise, which is random and uncorrelated across time or frequency, correlated noise exhibits patterns or structures, often arising in systems influenced by external factors or internal dynamics. This type of noise is significant in various fields, including signal processing, telecommunications, and financial modeling, as it can affect the performance and reliability of systems. Understanding correlated noise is essential for accurate modeling and analysis in these areas.

(2+2)/(2+2)=1

(2/2)+(2/2)=2

(2*2)-(2/2)=3

(2+2+2)-2=4

(2+2)+(2/2)=5

(2*2*2)-2=6

((2-.2)/.2)+2=7

2+2+2+2=8

(22/2)+2=9

((2+2)*2)+2=10

((2-.2)/.2)+2=11

(2+2+2)*2+=12

(22/2)+2=13

(2/.2)+(2+2)=14

2+(2/2)/.2=15

(2+2)*(2+2)=16

...??????????=17

((2+2)/.2)-2=18

((2+2)-.2)/.2=19

22- √(2+2)=20

230.
2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2 = 1073741824

21

As a fraction it is: 2/9

256

2, 128

2, 2, 64

2, 2, 2, 32

2, 2, 2, 2, 16

2, 2, 2, 2, 2, 8

2, 2, 2, 2, 2, 2, 4

2, 2, 2, 2, 2, 2, 2, 2

8192

32

2+2+2+2+2+2+2+2+2+2+2+2+(2x0)

=2+2+2+2+2+2+2+2+2+2+2+2+0

=24

2+2+2=6

2 * 2 * 2 * 2 * 2 * 2 * 2, or 27.

Broken down as follows.....

128

2, 64

2, 2, 32

2, 2, 2, 16

2, 2, 2, 2, 8

2, 2, 2, 2, 2, 4

2, 2, 2, 2, 2, 2, 2

128

2*64

2*2*32

2*2*2*16

2*2*2*2*8

2*2*2*2*2*4

2*2*2*2*2*2*2

2^7

2 + 2 - 2 = 2 or 2+(2-2)=2+0=2

128

2, 64

2, 2, 32

2, 2, 2, 16

2, 2, 2, 2, 8

2, 2, 2, 2, 2, 4

2, 2, 2, 2, 2, 2, 2

2 + 23 = 25

3 + 3 + 19 = 25

3 + 5 + 17 = 25

3 + 11 + 11 = 25

5 + 7 + 13 = 25

7 + 7 + 11 = 25

2 + 2 + 2 + 19 = 25

2 + 2 + 2 + 2 + 17 = 25

2 + 2 + 2 + 2 + 2 + 2 + 13 = 25

2 + 2 + 2 + 2 + 2 + 2 + 2 + 11 = 25

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 7 = 25

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 5 = 25

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 3 = 25

800

2, 400

2, 2, 200

2, 2, 2, 100

2, 2, 2, 2, 50

2, 2, 2, 2, 2, 25

2, 2, 2, 2, 2, 5, 5

480

2, 240

2, 2, 120

2, 2, 2, 60

2, 2, 2, 2, 30

2, 2, 2, 2, 2, 15

2, 2, 2, 2, 2, 3, 5

2+2+2+2+2+2+2+2+2+2+2+2

40

Its 4 ;p

-1+(2x9)=17

1 and 9 are not twos.

How about this: (2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2) /2 = 17

They are opposite parallel lines with points of (-2, 2) (2, 2) and (-2, -2) (2, -2)

210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 =1,024
1024

LCM factoring

512=2*2*2*2*2*2*2*2*2

648=2*2*2*2*3*3*3*3

lcm=2*2*2*2*2*2*2*2*2*3*3*3*3=41,472

The answer is 0 because at the end it is multiplied by 0

The expression "2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-3333333333333333333333333" consists of multiple subtractions of 2 followed by a long string of 3's. The repeated subtraction of 2 results in a negative number, specifically -36 (as there are 18 instances of 2), and the long string of 3's remains as is. Therefore, the expression simplifies to -36 followed by 3333333333333333333333333, but since they are not directly combined mathematically, the final result is more symbolic than numerical.

Negative numbers do not have prime factorizations.

2

you can break 800 in

2*2*2*2*2*5*5 = 2^5 * 5^2

so its divisible by any combination of multiplying numbers 2,2,2,2,2,5,5 where you can you only one in each combo.

so:

1

2

2*2=4

2*2*2=8

2*2*2*2=16

2*2*2*2*2=32

2*5=10

2*2*5=20

2*2*2*5=40

2*2*2*2*5=80

2*2*2*2*2*5=160

2*5*5=50

2*2*5*5=100

2*2*2*5*5=200

2*2*2*2*5*5=400

2*2*2*2*2*5*5=800

5

5*5=25

192

2, 96

2, 2, 48

2, 2, 2, 24

2, 2, 2, 2, 12

2, 2, 2, 2, 2, 6

2, 2, 2, 2, 2, 2, 3

2

It's not exactly true that people from The Netherlands specifically like to be called dutch, it's just that they consider it a fact of life that the English language has come up with that term. And it's considered somewhat funny that a small country like theirs has three uncorrelated words (Netherlands, Dutch and Holland) to identify it. Historically "Dutch" refers to "Diets", the German dialect that formed the basis for the language of the Netherlands, so there's no reason to dislike the word. Quite a few Dutch people know that Dutch also means other things in English, but then again in the Dutch language there are similar terms the other way around. For instance a "Dutch Treat" translates to "Amerkaans feestje" (American party).

Gcf

32=2*2*2*2*2

56=2*2*2*7

64=2*2*2*2*2*2

gcf=2*2*2 =8

2*2*2*2*2*2*2*2*2*2=1024

2^10 = 1024