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# Right and left quantiles

#### vasvet

##### New Member
Hi all,

I am struggling to prove left and right quantile relationship:

Right quantile (0.95) (x) = -left quantile (1-0.95) (-x)

I tried to prove with probability manipulations but struggle to get correct result.

Thank you

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @vasvet We use this a lot, in the case of the symmetrical standard normal distribution: N(-Z) = 1 - N(Z). For example, N(-2.33) = 1 - N(Z) = 1 - 99.0% = 1.0%. In Excel that's NORM.S.DIST(-2.33) = 1.0% = 1 - NORM.S.DIST(2.33) = 1 - 99.0%. Similarly, where N(-1) is the inverse standard normal CDF, N(-1)(p) = -N(-1)(1-p); for example, N(-1)(5%) = -N(-1)(95%) = -1.645. I'm thinking this is necessarily true of any symmetric distribution (but can't say for 100%); certainly symmetric is necessary condition. I wonder aloud if this is sufficient proof of symmetry, is this true (?): If F(-Z) = 1 - F(Z) --> skew = 0. I don't know ....

This video might be some help https://www.bionicturtle.com/forum/threads/t2-1-probability-functions-pdf-cdf-and-inverse-cdf.21449/ Re my notiation, N(.) is normal version of CDF and N^(-1) is an inverse CDF see https://www.bionicturtle.com/forum/...ulative-distribution-function.9438/post-41261 i.e.,
You definitely want to understand @ShaktiRathore 's above relationship. Miller's Chapter 2 does a decent job. We don't use the continuous pdf too much. The pdf integrates to the CDF, and we're arguably more interested in the relationships around the CDF, as Shakti illustrates. For example, if probability (p) = 5.0%, then:
• p = 5% = F(q), where F(.) is the cumulative distribution function, so if we are given the probability of 5% because we want a 95% confident normal VaR, then we use the inverse CDF to retrieve the quantile:
• q = F^(-1)(p) = F^(-1)(5%) = -1.645, where F^(-1) is the inverse CDF. "Inverse" implies that F[F^(-1)(p)] = p; e.g., F[F^(-1)(5%)] = F[-1.645] = 5%, just like also F^(-1)[F(q)] = q. This is why Dowd says "VaR is just a quantile;" i.e., because -1.645 is the quantile (q) retrieved by using the inverse CDF given a probability (p) of 5%. This is an essential FRM building block. I hope that adds something.

#### berrymucho

##### Member
Is "x" a random variable or a realization of a random variable (i.e. specific point)? If the former, then the relationship you quote is always true as the distribution of x is flipped like in a mirror, looking at the distribution of random variable x' = -x. If x is a specific point in the distribution of a ("fixed") random variable, see David's post.

Hi all,

I am struggling to prove left and right quantile relationship:

Right quantile (0.95) (x) = -left quantile (1-0.95) (-x)

I tried to prove with probability manipulations but struggle to get correct result.

Thank you

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