Basic Standard Normal Probability Question

This is a really basic question about a rule... I missed my lecture as I was studying for an exam, and was hoping that someone could shed some light on this for me.

(I'm making up symbols - <= is being 'less than or equal to')

I understand that probabilities have to be represented in a P(Z<= z) - but it looks as if you can assume that if P(Z<z) then P(Z<=z) - am I reading this right?

Thanks so much!!!

Well, I always write for ex. p<0.05 but I suppose that also means p<=0.05 . I only ever see the former usage though, unless I'm having a major memory lapse!!

breepi said:

This is a really basic question about a rule
I understand that probabilities have to be represented in a P(Z<= z) - but it looks as if you can assume that if P(Z<z) then P(Z<=z) - am I reading this right?

Click to expand...

The answer to this really depends upon the nature of the distribution.
If \(\displaystyle Z\) is continuous random variable then \(\displaystyle \mathcal{P}(Z<z)=\mathcal{P}(Z\le z)\).

However, that is not true for a random variable in general.

If \(\displaystyle F(z)=\mathcal{P}(Z\le z)\) is the cumulative distribution function for \(\displaystyle Z\)
then
\(\displaystyle \mathcal{P}(Z<z)=F(z-)=\displaystyle\lim _{x \to z^ - } F(x)\)
(the limit on the left at \(\displaystyle z\))