# Is this true?

EngWiPy
Hello,

Suppose X is a Chi-square random variable. Then what is:

$$\text{Pr}\left\{X<b\right\}$$?

Does the above probability is the CDF of X? The only difference is that there is no equality!

Thanks

EngWiPy
My question can we say that Pr[X<b] approximately equal Pr[X<=b]?

awkward
P(X < b) = P(X <= b) for continuous distributions, for example the chi-square distribution.

Staff Emeritus
My question can we say that Pr[X<b] approximately equal Pr[X<=b]?
Not approximately equal. Exactly equal.

The only time this isn't the case is with those non-continuous random variables for which P(x=b) can be non-zero for some values of b. This doesn't apply to the chi square distribution, which is an absolutely continuous probability distribution. "Absolutely continuous" essentially means it has a PDF; this a stronger constraint than merely being continuous.

EngWiPy
Thanks awkward and D_H, that is really relieving, since in my analysis I have Pr[X<b], and I was afraid it won't be correct to equate this with the CDF of Chi-square, i.e., Pr[X<=b], which has a closed form.

Thanks all