Probability with expectation and variance

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SUMMARY

The discussion focuses on calculating the expected value and variance of the distance D from a robot arm's soldering position on a motherboard, where the errors in positioning X and Y are independent and normally distributed with a mean of 0 and a standard deviation sigma. The joint probability density function (pdf) is expressed in polar coordinates as (r/sigma^2) * e^(-r^2/2sigma^2) * 1/2pi. The participants seek assistance in deriving the expected value E(D) and variance Var(D), as well as calculating E[|X^2 - Y^2|].

PREREQUISITES
  • Understanding of normal distribution and its properties
  • Familiarity with polar coordinates in probability
  • Knowledge of expectation and variance calculations
  • Basic concepts of joint probability density functions
NEXT STEPS
  • Learn how to derive the expected value from a joint probability density function
  • Study the properties of polar coordinates in multivariate distributions
  • Research methods for calculating variance in non-linear transformations
  • Explore advanced topics in probability theory, such as moment-generating functions
USEFUL FOR

Mathematicians, statisticians, and engineers involved in robotics and control systems, particularly those working with error analysis and probabilistic modeling.

chupi1289
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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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Please show your attempt.
 

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