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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.

akash
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  • They are refering to the cdf (cummulative density distribution). And the cdf of a normal distribution with zero mean is $0.5$ at $x=0$. – callculus42 Jul 08 '16 at 04:18

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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 $$F_W(0)=\Pr(W\le 0)=0.5.$$

André Nicolas
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  • Is w the mean ? – akash Jul 08 '16 at 04:22
  • $w$ is any number, a variable. – André Nicolas Jul 08 '16 at 04:24
  • @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 08 '16 at 04:38
  • Sir if mean has another value except 0 then what will be the cdf – akash Jul 08 '16 at 04:54
  • @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 08 '16 at 04:59
  • Sir I've asked if mean has its another value except zero. – akash Jul 08 '16 at 05:03
  • Then what will be its cdf.. – akash Jul 08 '16 at 05:04
  • If $W$ is normal with mean $\mu$ and standard deviation $\sigma$, then $F_W(w)=\Pr(W\le w)=\Pr\left(Z\le \frac{w-\mu}{\sigma}\right)$, where $Z$ is the standard normal. For any reasonable $z$, $\Pr(Z\le z)$ can be found by using a table of the standard normal. The cdf of $W$ can also be expressed as an integral, but for the normal that is not terribly computationally useful. – André Nicolas Jul 08 '16 at 05:09
  • @akash: About chat, I had a couple of negative experiences, so no longer use it. But I am perfectly willing to answer questions in comments, as long as the answers can be kept short. (Typing math in comments is difficult, since editing facilities are poor, and there are limits on length and layout.) – André Nicolas Jul 08 '16 at 05:11