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Q: Why is the standard normal probability distribution unique?

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A probability measure allocates a non-negative probability to each possible outcome. All individual probabilities together add up to 1. The "risk-neutral probability measure" is used in mathematical finance. Generally, risk-neutral probabilities are used for the arbitrage-free pricing of assets for which replication strategies exist. This is about relative pricing, based on possible replication strategies. The first argument is that a complete and arbitrage-free market setting is characterised by unique state prices. A state price is the price of a security which has a payoff of 1 unit only if a particular state is reached (these securities are called Arrow securities). In a complete market, every conceivable Arrow security can be traded. It is more easy to visualise these securities in terms of discrete scenarios. (On a continuous range of scenarios we would have to argue in terms of state price density.) The arbitrage-free price of every asset is the sum (over all scenarios) of the scenario-payoff weighted with its state price. Any pricing discrepancy with regards to an implicit state price would enable arbitrage in a complete market. The assumption is that the pursuit of such opportunities drives the prices towards the arbitrage-free levels. Hence the state prices are unique. Since the whole set of Arrow securities is the same as a risk-free bond (sure payoff of 1 unit at maturity), the price of the whole set of Arrow securities must be e^(-rt) (assuming we are now at maturity minus t). Risk-neutral probabilities can then be defined in terms of state prices, or vice versa. A probability measure has to fulfil the condition that the sum of all individual probabilities adds up to 1. Therefore, if we want to create an artificial probability distribution based on the state price distribution, we have to multiply each state price with e^(rt) in order to obtain its probability equivalent. It is not surprising then that any expectation taken under the risk-neutral probability measure grows at the risk-free rate. This is an artificial probability measure, why should we create such a construct? This connection allows us to exploit mathematical tools in probability theory for the purpose of arbitrage-free pricing. The main difficulty about risk-neutral probabilities is that the probability concepts used have not initially been developped for the purpose of financial pricing, therefore, two different languages are used, which can easily be confusing. The economic interpretation of a risk-neutral probability is a state price compounded at the risk-free rate. Anything that has an effect on a state price (preferences, real probability, ...), has an effect on the risk-neutral probability. So now we have a bridge to go from state prices to risk-neutral probabilities and back again. What is this good for? According to the second argument, we can, under certain conditions, specify the unique risk-neutral probability distribution of an underlying asset price with the help of an only incomplete specification of its real probability distribution, thanks to the Girsanov Theorem. If the innovation in the price of the underlying asset is driven by a Brownian motion, then all we need to obtain the risk-neutral probability distribution is the volatility parameter. What can we now do with this risk-neutral probability distribution? We can use the first argument to convert the obtained risk-neutral probability distribution back to a state price distribution, and the state price distribution applied to the payoff distribution (i.e. taking the sum over all scenarios) leads to the arbitrage-free price. These arguments save us a lot of trouble when trying to calculate the arbitrage-free price of an asset. They allow us to avoid the estimation of risk premia, by implicitly using those incorporated in the underlying asset price. The arbitrage-free price is, however, NOT independent of risk-premia. The price of the underlying asset is part of the pricing equation, and the risk-premia are inherent in this price, but because the price of the underlying asset is known to us, we obviously do not need estimate it. It is important to emphasise that the risk-neutral valuation approach only works if the asset to be priced can be perfectly replicated. This is often not true in reality, especially when dynamic replication strategies are involved. Paper explaining risk-neutral probabilities: http://ssrn.com/abstract=1395390

I believe you mean to say, equally probable. By stating they are separate events, I assume that they are independent and that there is a single unique outcome to each event that can be identified. Ok, then the chance of each event or outcome is 1/10.

A unique mode is the most occurring number in a set of data. For example, since the set: {1,2,2,4,4,5} does not have a single most occurring data value, the two modes (2 and 4) are averaged to make the mode value 3, even though it never occurs. This set does NOT have a unique mode. The set: {5,6,7,7,7,8,8} does have a unique mode, which is 7.

Vaa·shal - simply means Unique

The mode statistic is not always unique. For example, look at the following sequence of numbers: 1,2,2,3,4,4,5. Both 2 and 4 are modes for this sequence.

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The probability is zero! There is no such thing as "normal". Every child (and adult) has some unique characteristics and that makes them not normal - in that respect.

It is the desire to be different from all of the usual normal standards. It is the desire to be individual, and unique.

Yes

no

There is no personality considered to be normal. Everyone has unique personality traits and one person's personality cannot be considered more normal than anothers. The same with abnormal personalities. It depends on what standard you measure it against. Then you could qualify personality as normal or abnormal.

1. It does not have a unique origin Writing a school paper

odd unique strange abnormal

An example is warm-------cold standard-------unique dirty--------clean opposites. An example is warm-------cold standard-------unique dirty--------clean opposites.

no

Yes. Everyone's unique.

bland ordinary run of the mill standard

The meaning of normal practice is practices that are not customized. Normal practices don't have any unique features or services.

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