Probability distribution of function of continuous random variables

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SUMMARY

The discussion focuses on the probability distribution of the random variable Z, defined as the sum of a uniformly distributed variable X and a normally distributed variable Y. Specifically, X follows a Uniform distribution on the interval (A, B), while Y follows a Normal distribution with mean 0 and variance 1. Key questions include determining the probability density function (pdf) of Z, calculating probabilities such as P(Z < 3), and finding conditional probabilities and conditional pdfs related to Z and X.

PREREQUISITES
  • Understanding of Uniform distribution, specifically Uniform(A, B)
  • Knowledge of Normal distribution, particularly Normal(0, 1)
  • Familiarity with probability density functions (pdf)
  • Concepts of conditional probability and conditional pdf
NEXT STEPS
  • Study the convolution of probability distributions to derive the pdf of Z
  • Learn techniques for calculating probabilities involving continuous random variables
  • Explore conditional probability formulas and their applications in continuous distributions
  • Investigate numerical methods for evaluating integrals related to probability calculations
USEFUL FOR

Statisticians, data scientists, and students studying probability theory who are looking to deepen their understanding of functions of continuous random variables and their distributions.

kippers
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I hope someone can help me understand functions of random variables:

If X~Uniform(A,B), A < X < B
Y~Normal(0,1), -inf < Y < inf
and Z = X + Y

- what is the pdf of Z?
- how can I calculate a probability like P(Z < 3)?
- what is the conditional probability P(Z<z | X = x)?
- what is the conditional pdf of Z given X = x?

Any hint will be appreciated. Thank you.
 
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