# Value of cumulative distributed function at the origin

Consider a Gaussian distributed random variable with zero mean and standard deviation sigma. The value of its commulative distributed function at theorigin will be ....

In this question, 4 options are There .out of which 0.5 is correct ans.

Is Gaussian distribution same as normal distribution ? If yes then according to the nature of graph of normal distrubution its value should be zero.

• They are refering to the cdf (cummulative density distribution). And the cdf of a normal distribution with zero mean is $0.5$ at $x=0$. – callculus Jul 8 '16 at 4:18

Let $W$ be our Gaussian (normal) random variable. Then the cumulative distribution function $F_W(w)$ of $W$ is given by $$F_W(w)=\Pr(W\le w).$$ Note that $F_W(w)$ is the area under the bell-shaped curve from $-\infty$ to $w$.
Our normal has mean $0$, so has density function which is symmetric about $0$. Thus the area under the bell-shaped curve from $-\infty$ to $0$ is $0.5$, and therefore $$F_W(0)=\Pr(W\le 0)=0.5.$$
• $w$ is any number, a variable. – André Nicolas Jul 8 '16 at 4:24
• @akash: Maybe you thought that the cdf of $W$ at $0$ was the probability that $W$ is equal to $0$. The probability that $w=0$ is indeed $0$, since the normal has continuous distribution. But the cdf measures something else, the cdf at $w$ is the probability our random variable is less than or equal to $w$. – André Nicolas Jul 8 '16 at 4:38
• @akash: I do not understand the question, what is $p$? Maybe you could pose a specific question with specific numbers, then I can try to answer it. Make $\sigma$ specific too, since the answer may depend on it. (It didn't for the question you actually asked.) – André Nicolas Jul 8 '16 at 4:59