CDF of X: Probability Pr{X < b}

[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

 

[Bacle2]

I'm not sure I understand your question;

Have you checked the definitions, e.g:

http://en.wikipedia.org/wiki/Chi-squared_distribution

 

[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.

 

[D H]

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

 

[Bacle2]

Just a quick comment: one of the obstacles to assigning non-zero probability to

singletons is that an uncountable sum cannot converge unless only countably-many

terms are non-zero.