To calculate the probability of Z=Y^3 where Y is a standard normal distribution, one can approximate P(Z ≤ 1) by first determining P(Y ≤ 1), which is equivalent to P(X^3 ≤ 1) for X ~ N(0,1). The value for P(Y ≤ 1) is found to be approximately 0.84134 from standard normal distribution tables. For odd powers like Y^3, the appropriate root of the constant can be used for calculations. Additionally, to visualize the probability density function (pdf) for Y^3, one can plot the pdf of the normal distribution and adjust the X coordinates to Y^(1/3) for a linear representation in Y. The discussion emphasizes the relationship between transformations of normal variables and their probabilities.
