How to get probability from a normal distribution?

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SUMMARY

The discussion focuses on calculating the probability of a transformed variable Z, defined as Z=Y^3, where Y follows a standard normal distribution. The user correctly identifies that for odd powers, the probability can be approximated using the standard normal distribution values. Specifically, they conclude that P(Z ≤ 1) can be evaluated as P(Y ≤ 1), yielding a probability of approximately 0.84134. Additionally, the user inquires about the probability density function (pdf) of Y^3 and how to visualize it by transforming the coordinates.

PREREQUISITES
  • Understanding of standard normal distribution (N(0,1))
  • Knowledge of probability transformations
  • Familiarity with probability density functions (pdf)
  • Basic graphing skills for visualizing distributions
NEXT STEPS
  • Learn about probability transformations for non-linear functions
  • Explore the properties of the chi-square distribution
  • Study the graphical representation of transformed normal distributions
  • Investigate the implications of odd and even powers on probability calculations
USEFUL FOR

Statisticians, data analysts, and students studying probability theory who are interested in understanding transformations of normal distributions and their implications on probability calculations.

sneaky666
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If I had Z=Y^3 where Y is a standard normal distribution. How would I approx. calculate the probability of Z<=1 ?, I would understand it if it was Z=Y^2 which is chi-square...
 
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P(Z ≤ 1)=P(Y ≤ 1). In general for odd powers, you just need to take the appropriate root of the constant of interest.
 
well i have to do P(Y<=1) = P(X^3<=1) = P(X<=1), and since X ~N(0,1), so I can just look in the book for the standard normal distribution values
which is just 0.84134, is that right?
By the way out of curiosity how would the pdf graph look for Y^3 ?
 
I assume you meant X^3 = Y. Plot pdf for normal distribution. Change X coordinates to Y^(1/3) and replot to be linear in Y.
 

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