How to get probability from a normal distribution?

[sneaky666]

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...

 

[mathman]

P(Z ≤ 1)=P(Y ≤ 1). In general for odd powers, you just need to take the appropriate root of the constant of interest.

 

[sneaky666]

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 ?

 

[mathman]

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.

 

[blue_raver22]

To calculate the probability of Z<=1, we can use the cumulative distribution function (CDF) of a standard normal distribution. The CDF represents the probability that a random variable falls below a certain value.

In this case, we can use the CDF of Y to calculate the probability of Z<=1. Since Y is a standard normal distribution, we can use a table or a statistical software to find the CDF value for Y=1. This value represents the probability that Y is less than or equal to 1.

Next, we can use the formula P(Z<=1) = P(Y^3<=1) = P(Y<=1^(1/3)) = P(Y<=1), since 1^(1/3) = 1. This means that the probability of Z<=1 is equal to the probability of Y<=1, which we calculated in the previous step.

Therefore, to approximate the probability of Z<=1, we can use the CDF value for Y=1 that we found earlier. This approach can be used for any value of Z that is a function of Y, as long as Y is a standard normal distribution.