Expectation and Variance

In summary, we can find E(Y) and Var(Y) by using the properties of expected value and variance for independent random variables.
  • #1
bsmith2000
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Homework Statement



Say X has a density f(x) = 3x^(-4) if x > 1, and 0 otherwise. Now say X1,...,X16 are independent with density f. Let Y = (X1X2...X16)^(1/16). Find E(Y) and Var(Y).

Homework Equations



Var(Z) = E(Z^2) - [E(Z)]^2
E(Z) = Integral from -inf to +inf of z*f(z)dz

The Attempt at a Solution



I found E(X) = 3/2 and Var(X) = 3/4, using the above formulas. However, I still do not even know how to approach the actual question of the problem.

Thank you all in advance!
 
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  • #2


Hello, thank you for your question. To find E(Y) and Var(Y), we can use the property of expected value and variance for independent random variables. Since X1,...,X16 are independent, we can write E(Y) and Var(Y) as follows:

E(Y) = E(X1)E(X2)...E(X16)^(1/16) = (3/2)^16^(1/16) = 1.5

Var(Y) = Var(X1)Var(X2)...Var(X16)^(1/16) = (3/4)^16^(1/16) = 0.75

Alternatively, we can also use the formula for variance in terms of expected value: Var(Y) = E(Y^2) - [E(Y)]^2.

To find E(Y^2), we can use the property of independent random variables again: E(Y^2) = E(X1^2)E(X2^2)...E(X16^2)^(1/16) = (3/2)^16^(1/16) = 1.5^2 = 2.25.

Therefore, Var(Y) = E(Y^2) - [E(Y)]^2 = 2.25 - (1.5)^2 = 0.75.

I hope this helps. Let me know if you have any further questions.
 

1. What is expectation in statistics?

The expectation or mean of a random variable is the average value that we expect to see if we repeat an experiment many times. It is calculated by multiplying each possible outcome by its probability and summing them up.

2. How is variance defined?

Variance is a measure of how much the values of a random variable differ from the mean. It is calculated by taking the average of the squared differences between each value and the mean.

3. Why is expectation and variance important in probability distributions?

Expectation and variance provide important information about the shape and spread of a probability distribution. They help us understand the central tendency and variability of a random variable, which is useful for making predictions and drawing conclusions from data.

4. How are expectation and variance related?

The variance of a random variable is the average squared distance from the mean. Therefore, it is related to the expectation by the formula Var(X) = E[(X - E[X])^2]. In other words, the variance is the expectation of the squared difference between each value and the mean.

5. What are some real-world applications of expectation and variance?

Expectation and variance are used in many fields, such as finance, engineering, and physics. In finance, they are used to model stock prices and risk. In engineering, they are used to analyze the reliability of systems. In physics, they are used to describe the behavior of particles in quantum mechanics.

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