Probability with expectation and variance

In summary, the conversation discusses the use of a robot arm for soldering on a motherboard and how to determine the magnitude of errors in its location. The errors are assumed to have a normal distribution with mean 0 and a standard deviation of sigma. The density function of the distance D is given by D = SQRT(X^2 + Y^2). The expected value and variance of D can be calculated as E(D) = 0 and Var(D) = 2sigma^2. Additionally, the conversation mentions the calculation of E[|X^2 - Y^2|] using a joint probability density function in polar coordinates.
  • #1
chupi1289
1
0
A robot arm solders a component on a motherboard. The arm has
small tiny errors when locating the correct place on the board. This
exercise tries to determine the magnitude of the error so that we know
the physical limitations for the size of the component connections. Let
us say that the right place to be soldered is the origin (0,0), and the
actual location the arm goes to is (X,Y ). We assume that the errors
X and Y are independent and have the normal distribution with mean
0 and a certain standard deviation sigma.
(a) What is the density function of the distance
D = SQRT ( X^2 + Y^2)


(b) Calculate its expected value and variance:
E(D) and Var(D)


(c) Calculate
E[|X^2 - Y^2|]

Ok so I changed to polar and have my joint pdf as follows:

(r/sigma^2) * e^(-r^2/2sigma^2) *1/2pi

Don't know how to calculate expectation and variance. I think I'm doing it wrong
 
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  • #2
Please show your attempt.
 

FAQ: Probability with expectation and variance

What is probability?

Probability is a way of measuring the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 represents impossibility and 1 represents certainty.

What is expectation?

Expectation, also known as the expected value, is a way of measuring the average outcome of a random experiment. It is calculated by multiplying each possible outcome by its probability and then adding all of these values together.

What is variance?

Variance is a measure of how spread out a set of data is. In the context of probability, it is a measure of how much the actual outcomes of a random experiment differ from the expected value. It is calculated by taking the square of the difference between each outcome and the expected value, multiplying by its probability, and then adding all of these values together.

How are expectation and variance related?

Expectation and variance are both measures of central tendency for a set of data. The expectation represents the average outcome, while the variance measures how much the actual outcomes deviate from this average. In other words, the expectation and variance work together to provide a more complete picture of the data.

What are some real-world applications of probability with expectation and variance?

Probability with expectation and variance is used in a wide range of fields, including finance, economics, insurance, and gambling. It can be used to make predictions, evaluate risks, and make informed decisions based on uncertain outcomes.

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