Likelihood of Winning - Probability Density FunctionIn developing the "Likelihood of Winning - Probability Density Function (PDF)" indicator, my aim was to offer traders a statistical tool to quantify the probability of reaching target prices. This indicator, grounded in risk assessment principles, enables users to analyze potential outcomes based on the normal distribution, providing insights into market dynamics.
The tool's flexibility allows for customization of the data series, lookback periods, and target settings for both long and short scenarios. It features a color-coded visualization to easily distinguish between probabilities of hitting specified targets, enhancing decision-making in trading strategies.
I'm excited to share this indicator with the trading community, hoping it will enhance data-driven decision-making and offer a deeper understanding of market risks and opportunities. My goal is to continuously improve this tool based on user feedback and market evolution, contributing to more informed trading practices.
This indicator leverages the "NormalDistributionFunctions" library, enabling easy integration into other indicators or strategies. Users can readily embed advanced statistical analysis into their trading tools, fostering innovation within the Pine Script community.

# Normaldistribution

NormalDistributionFunctionsLibrary "NormalDistributionFunctions"
The NormalDistributionFunctions library encompasses a comprehensive suite of statistical tools for financial market analysis. It provides functions to calculate essential statistical measures such as mean, standard deviation, skewness, and kurtosis, alongside advanced functionalities for computing the probability density function (PDF), cumulative distribution function (CDF), Z-score, and confidence intervals. This library is designed to assist in the assessment of market volatility, distribution characteristics of asset returns, and risk management calculations, making it an invaluable resource for traders and financial analysts.
meanAndStdDev(source, length)
Calculates and returns the mean and standard deviation for a given data series over a specified period.
Parameters:
source (float) : float: The data series to analyze.
length (int) : int: The lookback period for the calculation.
Returns: Returns an array where the first element is the mean and the second element is the standard deviation of the data series for the given period.
skewness(source, mean, stdDev, length)
Calculates and returns skewness for a given data series over a specified period.
Parameters:
source (float) : float: The data series to analyze.
mean (float) : float: The mean of the distribution.
stdDev (float) : float: The standard deviation of the distribution.
length (int) : int: The lookback period for the calculation.
Returns: Returns skewness value
kurtosis(source, mean, stdDev, length)
Calculates and returns kurtosis for a given data series over a specified period.
Parameters:
source (float) : float: The data series to analyze.
mean (float) : float: The mean of the distribution.
stdDev (float) : float: The standard deviation of the distribution.
length (int) : int: The lookback period for the calculation.
Returns: Returns kurtosis value
pdf(x, mean, stdDev)
pdf: Calculates the probability density function for a given value within a normal distribution.
Parameters:
x (float) : float: The value to evaluate the PDF at.
mean (float) : float: The mean of the distribution.
stdDev (float) : float: The standard deviation of the distribution.
Returns: Returns the probability density function value for x.
cdf(x, mean, stdDev)
cdf: Calculates the cumulative distribution function for a given value within a normal distribution.
Parameters:
x (float) : float: The value to evaluate the CDF at.
mean (float) : float: The mean of the distribution.
stdDev (float) : float: The standard deviation of the distribution.
Returns: Returns the cumulative distribution function value for x.
confidenceInterval(mean, stdDev, size, confidenceLevel)
Calculates the confidence interval for a data series mean.
Parameters:
mean (float) : float: The mean of the data series.
stdDev (float) : float: The standard deviation of the data series.
size (int) : int: The sample size.
confidenceLevel (float) : float: The confidence level (e.g., 0.95 for 95% confidence).
Returns: Returns the lower and upper bounds of the confidence interval.

Normal Distribution Asymmetry & Volatility ZonesNormal Distribution Asymmetry & Volatility Zones Indicator provides insights into the skewness of a price distribution and identifies potential volatility zones in the market. The indicator calculates the skewness coefficient, indicating the asymmetry of the price distribution, and combines it with a measure of volatility to define buy and sell zones.
The key features of this indicator include :
Skewness Calculation : It calculates the skewness coefficient, a statistical measure that reveals whether the price distribution is skewed to the left (negative skewness) or right (positive skewness).
Volatility Zones : Based on the skewness and a user-defined volatility threshold, the indicator identifies buy and sell zones where potential price movements may occur. Buy zones are marked when skewness is negative and prices are below a volatility threshold. Sell zones are marked when skewness is positive and prices are above the threshold.
Signal Source Selection : Traders can select the source of price data for analysis, allowing flexibility in their trading strategy.
Customizable Parameters : Users can adjust the length of the distribution, the volatility threshold, and other parameters to tailor the indicator to their specific trading preferences and market conditions.
Visual Signals : Buy and sell zones are visually displayed on the chart, making it easy to identify potential trade opportunities.
Background Color : The indicator changes the background color of the chart to highlight significant zones, providing a clear visual cue for traders.
By combining skewness analysis and volatility thresholds, this indicator offers traders a unique perspective on potential market movements, helping them make informed trading decisions. Please note that trading involves risks, and this indicator should be used in conjunction with other analysis and risk management techniques.

Oscillator: Which follows Normal Distribution?When doing machine learning using oscillators, it would be better if the oscillators were normally distributed.
So I analyzed the distribution of oscillators.
The value of the oscillator was divided into 50 groups each from 0 to 100.
ex) if rsi value is 45.43 -> group_44, 58.23 -> group_58
Ocscillators : RSI, Stoch, MFI, WT, RVI, etc....
Caution: The normal distribution was verified through an empirical formula.

RSI is in Normal Distribution?Does RSI Follow a Normal Distribution?
The value of RSI was converted to a value between 0~2, 2~4, ..., 98~100, and the number of samples was graphed.
The Z values are expressed so that the values corresponding to 30 and 70 of the RSI can be compared with the standard normal distribution.
Additionally, when using the RSI period correction function of the 'RSI Candle Advanced V2' indicator that I made before, it shows no change in standard deviation.
RSI는 정규분포를 따를까요
RSI의 값을 0~2, 2~4, ..., 98~100 사이 값으로 변환하고 그 표본 갯수를 그래프로 표현하였습니다.
Z 값은 RSI의 30, 70에 해당하는 값을 표준정규분포와 비교할 수 있도록 표현하였습니다.
추가적으로 제가 예전에 만들었던 'RSI Candle Advanced V2' 지표의 RSI 기간 보정 함수를 사용할 경우 표준편차의 변화가 없음을 보입니다.

Return Abnormality Score [SpiritualHealer117]The Return Abnormality Score indicator is designed to help traders identify potential reversals in price by detecting abnormal daily returns beyond a certain significance level. The indicator uses a normal cumulative distribution function to calculate the probability of the daily return and flags it when it exceeds the specified significance level.
Traders can use this indicator by monitoring the abnormality score. If the daily return is negative, the probability is multiplied by a negative number. Therefore, if the abnormality score goes above the positive threshold, it suggests that the price is oversold, while if it goes below the negative threshold, it indicates that the price is overbought. It can also be helpful for spotting bear or bull traps due to their irregular behavior.
Depending on the trader's preference, the indicator can be smoothed or unsmoothed.
This indicator should be paired with other technical analysis tools like SSL Hybrid for trend confirmation, and proper risk management strategies.

normsinvLibrary "normsinv"
Description:
Returns the inverse of the standard normal cumulative distribution.
The distribution has a mean of zero and a standard deviation of one; i.e.,
normsinv seeks that value z such that a normal distribtuion of mean of zero
and standard deviation one is equal to the input probability.
Reference:
github.com
normsinv(y0)
Returns the inverse of the standard normal cumulative distribution. The distribution has a mean of zero and a standard deviation of one.
Parameters:
y0 : float, probability corresponding to the normal distribution.
Returns: float, z-score

cndevLibrary "cndev"
This function returns the inverse of cumulative normal distribution function
Reference:
The Full Monte, by Boris Moro, Union Bank of Switzerland . RISK 1995(2)
CNDEV(U)
Returns the inverse of cumulative normal distribution function
Parameters:
U : float,
Returns: float.

ctndLibrary "ctnd"
Description:
Double precision algorithm to compute the cumulative trivariate normal distribution
found in A.Genz, Numerical computation of rectangular bivariate and trivariate normal
and t probabilities”, Statistics and Computing, 14, (3), 2004. The cumulative trivariate
normal is needed to price window barrier options, see G.F. Armstrong, Valuation formulae
or window barrier options”, Applied Mathematical Finance, 8, 2001.
References:
link.springer.com
www.tandfonline.com
citeseerx.ist.psu.edu
The Complete Guide to Option Pricing Formulas, 2nd ed. (Espen Gaarder Haug)
CTND(LIMIT1, LIMIT2, LIMIT3, SIGMA1, SIGMA2, SIGMA3)
Returns the Cumulative Trivariate Normal Distribution
Parameters:
LIMIT1 : float,
LIMIT2 : float,
LIMIT3 : float,
SIGMA1 : float,
SIGMA2 : float,
SIGMA3 : float,
Returns: float.

norminvLibrary "norminv"
Description:
An inverse normal distribution is a way to work backwards
from a known probability to find an x-value. It is an informal term and
doesn't refer to a particular probability distribution. Returns the
value of the inverse normal distribution function for a specified value,
mean, and standard deviation.
Reference:
github.com
support.microsoft.com
norminv(x, mean, stdev)
Returns the value of the inverse normal distribution function for a specified value, mean, and standard deviation.
Parameters:
x : float, The input to the normal distribution function.
mean : float, The mean (mu) of the normal distribution function
stdev : float, The standard deviation (sigma) of the normal distribution function.
Returns: float.

cndLibrary "cnd"
Cumulative Normal Distribution
CND1(x)
Returns the Cumulative Normal Distribution (CND) using the Hart (1968) method. (preferred method, 14-18 decimal accuracy)
Parameters:
x : float,
Returns: float.
CND2(x)
Returns the Cumulative Normal Distribution (CND) using the Abromowitz and Stegun (1974) Polynomial Approximation.
Parameters:
x : float,
Returns: float.
CND3(x)
Returns the Cumulative Normal Distribution (CND) using Newton-Cotes method, Boole’s rule
Parameters:
x : float,
Returns: float.

One-Sided Gaussian Filter w/ Channels [Loxx]One-Sided Gaussian Filter w/ Channels is a Gaussian Moving Average that is calculated using a Fibonacci weighting function. Keltner channels have been added to show zones of exhaustion. A better name would be "Half Gaussian bell weighted" or "Half normal distribution weighted" indicator, since the weights for calculation of the average (similar to linear weighted average) are taken from a normal distribution curve like function--but only the half of the curve is used for calculation.
Information of the Gaussian distribution can be found here : en.wikipedia.org and once you take a look at the standard normal distribution curve, it will be much clearer what is exactly done in this indicator.
After the Gaussian Filter is applied to the source input, an Ehlers' 2-Pole Super Smoother is applied to reduce noise without significant lag.
Included:
Bar coloring
Signals
Alerts
Loxx's Expanded Source Types

Is the Bollinger Bands assumption wrong?Bollinger Bands are the result of the assumption that closing prices will follow a normal distribution.
However, when I actually calculated the probability, the closing price does not follow a normal distribution.
According to the normal distribution, the probability that Z > 2 should be 2.2%, but on the chart, the probability is 6~9%.
Can we get a useful value for Bollinger Bands that we can use in our strategy?
We can measure volatility, but can we judge volatility based on a fixed value?
To the right of each Bollinger band value, the probability that the price is above the band is displayed.
The script is simply annotated with how each probability is calculated.

Probability Distribution HistogramProbability Distribution Histogram
During data exploration it is often useful to plot the distribution of the data one is exploring. This indicator plots the distribution of data between different bins.
Essentially, what we do is we look at the min and max of the entire data set to determine its range. When we have the range of the data, we decide how many bins we want to divide this range into, so that the more bins we get, the smaller the range (a.k.a. width) for each bin becomes. We then place each data point in its corresponding bin, to see how many of the data points end up in each bin. For instance, if we have a data set where the smallest number is 5 and the biggest number is 105, we get a range of 100. If we then decide on 20 bins, each bin will have a width of 5. So the left-most bin would therefore correspond to values between 5 and 10, and the bin to the right would correspond to values between 10 and 15, and so on.
Once we have distributed all the data points into their corresponding bins, we compare the count in each bin to the total number of data points, to get a percentage of the total for each bin. So if we have 100 data points, and the left-most bin has 2 data points in it, that would equal 2%. This is also known as probability mass (or well, an approximation of it at least, since we're dealing with a bin, and not an exact number).
Usage
This is not an indicator that will give you any trading signals. This indicator is made to help you examine data. It can take any input you give it and plot how that data is distributed.
The indicator can transform the data in a few ways to help you get the most out of your data exploration. For instance, it is usually more accurate to use logarithmic data than raw data, so there is an option to transform the data using the natural logarithmic function. There is also an option to transform the data into %-Change form or by using data differencing.
Another option that the indicator has is the ability to trim data from the data set before plotting the distribution. This can help if you know there are outliers that are made up of corrupted data or data that is not relevant to your research.
I also included the option to plot the normal distribution as well, for comparison. This can be useful when the data is made up of residuals from a prediction model, to see if the residuals seem to be normally distributed or not.

ema exhaustion (exa)The exa is an oscillator that combines fisher transform with distance from moving average and it is based on a theory that exhaustion can be derived from how far price is able to extend from a moving average, on average.
The fisher transform converts price into a gaussian normal distribution, also known as a bell curve {1}. A normal distribution is a type of probability distribution for a real-valued random variable {2}. Applying this method to the price of an asset can help to identify probabilities, but it will never identify certainties.
‘exa’ is an abbreviation for ema exhaustion. It can be used to identify when price is probable to revert to the mean but I prefer using it to confirm entries that are signaled following a reversion to the mean (aka buying the dip in bull markets). When price gets oversold into support, in a bull trend, then that can provide a good opportunity to enter long. However that isn’t necessarily the case when the same metrics indicate oversold conditions in a bear trend. In this situation the exa is best suited for identifying profit taking opportunities on shorts.
The default settings are a 9 lookback period and a 50 ema. By default signals will be derived from how far price is from the 50 ema relative to the probable distribution of the last 9 periods. If the exa is above 2, or below -2, then the price is in the 80th percentile of the prior 9 candles. Being outside of 3, or -3, represents the 90th percentile and 4, or -4, represents the 95th percentile.
Those ranges will never indicate a necessity of reverting to the mean, but they will indicate a higher and higher probability. I prefer to use this oscillator in combination with an indicator(s) that identifies the trend. When the oscillator reaches -2 in a bull trend then it can confirm long entry signals, whereas if it reaches +2 in a bull trend then it can be used to confirm signals to take profit.
Crossovers are especially significant because they indicate a shift in the tide. When the exa reaches 2 without crossing over then it is very much in a position to move to 3 or 4+. When it crosses above 2 then it is an indication that price is extended from the mean and exhausted.
This is certainly not a situation that implies price will revert to the mean, it simply provides confirmation.
The default settings are what I have been finding most effective personally, however that is mostly a function of the trend following tools that I use. The same principles should apply with all settings and I would encourage users to experiment with various lookback periods and emas.
{1} www.investopedia.com
{2} en.wikipedia.org

Simplest volatility bandsVolatility bands based on average candle percentage spread. Tested on BTCUSD charts only.
Based on the 68-95-99.7 rule, it seems that the spread, for daily and 4-H candles, follows a normal distribution: that means, around 85% of candles have a %-spread within sma(low/high, some_len) and sma(high/low, some_len) , and around 95% of candles within the pow2 of that range.
If you take the mean between the boundaries of the first %-spreads band, and calculate the 1.5 standard deviation of past some_len candles (I'm speaking from memory, it has been a while since I did them), the 1.5 standard deviation bands match similarly the %-spread bands, and around 85% of the candles are within these %-spread bands.
If you then take the pow2 of the bands, it will be similar to the 2 * std of the original bands, with around 95% of data within the pow2 bands.
You can take ema or other similar means with similar results, and the same for different lengths, but it seems that sma with a len of 14 is the more stable ones for both daily and 4-H, and taken other average calculations doesn't cause too many differences respect to the sma. I haven't tested too much for lower or higher timeframes.
With those %-spread bands, I multiple and divide those spreads to the open value of a new candle to get the two bands.
So, in short, you know that 85% of candles are within the closer bands, and around 95% of candles, around the bigger one. Once a new candle is born, the bands won't move (the bands are calculated from the previous candle, so the current candle's price movement doesn't move the band).
Going out the bands implies a sudden increase in volality, which usually causes rejection. They happen mostly at breakouts and ends of heavy trends. If a candle closes above the bigger band, you have probably got a breakout (a rejection rarely happens if the candle have already closed), although a breakout can happen without closing above the bands if volatility was already high.
If a trend is already stablished and is healthy, you won't probably see candles going out the bands, not even with a wick. When the trend is parabolic, and goes above the candle, the trend has probably ended, although the trend can be exhausted without going out the bands as well.
Heavy but not yet exhausted trends (specially recently started heavy downtrends), usually reach the bottom of the bigger bands during 4 o 5 contiguous candles (check visually looking at bitcoin history though, I'm speaking from memory).
So, the possibilities are multiple and you cannot use the bands to form a strategy, as usual. It can be comfortable enough psycologically for going to sleep, by moving your stop-loss to a point out of the bands in the opposite direction of your trade, and adjusting your position size accordingly; or just to check momentum looking at how close are the candle limits to the bands.
But, as usual, you are responsible of what you do with your money :)