# What is the linearity of expectation?

## What is the linearity of expectation?

Linearity of expectation is the property that the expected value of the sum of random variables is equal to the sum of their individual expected values, regardless of whether they are independent. The expected value of a random variable is essentially a weighted average of possible outcomes.

## Is expectation always linear?

The definition of expectation follows our intuition. In particular, the following theorem shows that expectation preserves the inequality and is a linear operator. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. 1.

Does linearity of expectation hold for dependent variables?

Linearity of expectation holds for both dependent and independent events. On the other hand the rule E[R1R2] = E[R1]*E[R2] is true only for independent events.

What are the properties of expectation?

The following properties of expectation apply to discrete, continuous, and mixed random variables:

• Indicator function. The expectation of the indicator function is a probability: (5.56)
• Linearity. Expectation is a linear operator: (5.58)
• Nonnegative.
• Symmetry.
• Independence.

### Which is true about the linearity of expectation?

Some interesting facts about Linearly of Expectation: Linearity of expectation holds for both dependent and independent events. On the other hand the rule E [R 1 R 2] = E [R 1 ]*E [R 2] is true only for independent events. Linearity of expectation holds for any number of random variables on some probability space.

### Why does my ELISA have poor dilution linearity?

Assays for multiple analytes such as our HCP ELISA will often show a lack of dilution linearity for certain samples. In the case of HCP assays this lack of dilution linearity is usually due to insufficient excess of antibody for some of the HCPs found in your sample.

Which is an example of poor dilution linearity?

Poor Dilution Linearity Demonstration of dilution linearity (also termed dilution parallelism or dilution recovery) of samples containing the analyte of interest is a critical experiment to validate the specificity and accuracy of a given method.

Is the proof of linearity of expectation the same for continuous random variables?

For continuous random variables, the proof is essentially the same except that the summations are replaced by integrals. Additionally, it is easy to extend the proof for two random variables to the more general case using the properties of expected value. □ _\\square □ ​