# Probability with Chi-Square distribution

What is the difference, when calculating probabilities of Chi-Square distributions, between $<$ and $\leq$ or $>$ and $\geq$.

For example, say you are asked to find P$(\chi_{5}^{2} \leq 1.145)$.

I know that this is $=0.05$ from the table of Chi-Square distributions, but what if you were asked to find P$(\chi_{5}^{2} < 1.145)$? How would this be different?

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There is no numerical difference since $P\{\chi_5^2 = 1.145\} = 0$ and so $$P\{\chi_5^2 < 1.145\} = P\{\chi_5^2 \leq 1.145\} - P\{\chi_5^2 = 1.145\} = P\{\chi_5^2 \leq 1.145\}.$$ Indeed, $P\{\chi_5^2 = a\} = 0$ for all real numbers $a$. – Dilip Sarwate Feb 7 '13 at 23:56
If you know calculus, you can use the fact that $$\Pr[a \le X \le b] = \int_a^b f(x) \mathrm dx$$ Consider $$\Pr[a \le X \le a] = \int_a^a f(x) \mathrm dx = 0$$ – George V. Williams Feb 8 '13 at 0:05

The $\chi^2$ distributions are continuous distributions. If $X$ has continuous distribution, then $$\Pr(X\lt a)=\Pr(X\le a).$$ If $a$ is any point, then $\Pr(X=a)=0$. So in your case, the probabilities would be exactly the same.
If $X$ is a random variable with density function $f(x)$, then $\Pr(X=a)=\int_a^af(x)\,dx$. The integral from $a$ to $a$ of any function is $0$. Or, in terms of areas, recall that the probability that $X$ lies between $a$ and $b$ is the area under the density function, from $a$ to $b$. Well, the area from $a$ all the way up to $a$ is $0$. – André Nicolas Feb 8 '13 at 0:07