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****Probabilistic Tool for Freezing
Temperatures**

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A ** cumulative
distribution function** or

**F(x)
= Pr{X <= x} = ****∫**_{X<=x }**f(x)dx**

Therefore the CDF specifies
probabilities that the random quantity X will not exceed specific values. So in the case of surface temperatures,

The variable x = 32F and X = the NDFD
temperature field.

Since we don’t know f(x) or
the PDF for the temperature forecast, we assume a normalized or Gaussian
distribution, where temperature (T) is the mean and the variance or standard
deviation (σ) is the ensemble mean. Thus we can compute a ** cumulative
normal distribution function**.

**Prob**** (T < 32) = 1/σ _{T }√2π **

The probability of
temperatures reaching 32 degrees F is derived from the computed cumulative
normal distribution of the 1200 UTC NDFD temperature grid data (T_{mean}) and the 2-meter temperature grid spread
data (**σ _{T}** ) from the GFS ensemble (GEFS), and setting 32F as the
limit for the integration.

This data from for this tool is generated once per day at approximately 1400 UTC for the entire temporal domain of the NDFD data in 6-hour increments extending out through the 156 hour forecast.

*References*

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*Wilks**, Daniel S., Statistical Methods in the Atmospheric
Sciences, 2 ^{nd} Edition, 2006.*