Worked Tail-Probability Calculations Across Families

From a Threshold to a Probability

A tail-probability calculation starts with a model and a clearly stated event. For example, “exceeding 10” means \(P(X>10)\), whereas “at most 10” means \(P(X\leq 10)\). These events are complements, but the formula used to calculate each probability depends on the distribution family. A sound calculation also checks that a threshold and the model’s scale parameters use compatible units.

Write \(F(x)=P(X\leq x)\) for the cumulative distribution function and \(S(x)=P(X>x)\) for the survival function. A continuous distribution assigns probability zero to any single point, so its probabilities of \(X>x\) and \(X\geq x\) agree. This tutorial concentrates on evaluating such probabilities from specified models, rather than on identifying a family from a graph or comparing general tail behavior.

Definition: The upper-tail probability at threshold \(t\) is \(P(X>t)=S(t)\). The lower-tail probability is \(P(X\leq t)=F(t)\). For every random variable, \(S(t)=1-F(t)\). For continuous distributions, \(P(X\geq t)=P(X>t)\), and \(P(X<t)=P(X\leq t)\).

Formulas and Parameter Units

The table collects the formulas needed for the calculations below. Here \(\Phi\) is the standard normal CDF. Parameters are assumed to be in the families’ usual ranges, and the lognormal model is stated using an explicit reference unit.

Family and parameters	Upper-tail formula	Unit check
Normal \(X\sim N(\mu,\sigma^2)\), \(\sigma>0\)	\(1-\Phi((t-\mu)/\sigma)\)	\(\mu,\sigma,t\) have the same units
Logistic location \(\mu\), scale \(s>0\)	\(1/[1+\exp((t-\mu)/s)]\)	\(\mu,s,t\) have the same units
Laplace location \(\mu\), scale \(b>0\)	\(\frac12\exp(-(t-\mu)/b)\), for \(t\geq\mu\)	\(\mu,b,t\) have the same units
Lognormal: \(\log(X/u)\sim N(\mu,\sigma^2)\), \(X>0\)	\(1-\Phi((\log(t/u)-\mu)/\sigma)\), for \(t>0\)	\(u,t\) are positive quantities in the same units; \(\mu,\sigma\) are dimensionless
Weibull shape \(k>0\), scale \(\lambda>0\)	\(\exp[-(t/\lambda)^k]\), for \(t\geq0\)	\(t,\lambda\) have the same units; \(k\) is dimensionless
Pareto Type I minimum \(x_m>0\), shape \(\alpha>0\)	\((x_m/t)^\alpha\), for \(t\geq x_m\)	\(t,x_m\) have the same units; \(\alpha\) is dimensionless

In the lognormal row, the parentheses in the normal-standardization are important: the standardized value is \((\log(t/u)-\mu)/\sigma\). The Pareto formula applies only on the stated upper support; if \(t<x_m\), then \(P(X>t)=1\). Similarly, a nonnegative Weibull variable has \(P(X>t)=1\) for \(t<0\). A threshold outside the support should be handled from the support, not by extending a formula beyond where it applies.

Result: For every random variable and every real threshold \(t\), \(P(X>t)=1-F(t)\). If \(X\) has a continuous distribution, then also \(P(X\geq t)=1-F(t)\). For a distribution with a point mass at \(t\), \(P(X\geq t)\) instead includes that mass.

To verify the first identity, the events \(\{X>t\}\) and \(\{X\leq t\}\) are disjoint and their union is the whole sample space. Therefore their probabilities sum to one. For a continuous distribution, \(P(X=t)=0\), so \(P(X\geq t)=P(X>t)+P(X=t)=P(X>t)\). This distinction matters when interpreting a threshold event: complementing \(F(t)\) gives the strict upper tail exactly, even without continuity.

Worked Comparisons for Normal, Logistic, and Laplace Tails

For a normal model, first standardize the threshold. For logistic and Laplace models, substitute the threshold into the family’s survival formula, taking care to use the family’s own scale parameter. The same numerical value for two different scale parameters does not make the models equivalent.

Worked Example: Comparing Three Upper Tails at Six Units

Let \(X_N\sim N(0,3^2)\), \(X_L\) have a logistic distribution with location \(0\) and scale \(2\), and \(X_A\) have a Laplace distribution with location \(0\) and scale \(3\). Calculate the probability of exceeding \(6\) in each model.

Normal. The standardized threshold is \(z=(6-0)/3=2\). Thus

P(X_N>6)=1-\Phi(2)\approx 1-0.97725=0.02275.

Logistic. With scale \(s=2\), the exponent’s argument is \((6-0)/2=3\). Therefore

P(X_L>6)=\frac{1}{1+\exp(3)} \approx \frac{1}{1+20.0855} \approx 0.04743.

Laplace. Since \(6\geq 0\), use the upper-tail branch with \(b=3\):

P(X_A>6)=\frac12\exp(-(6-0)/3) =\frac12\exp(-2) \approx \frac12(0.13534) \approx 0.06767.

At this particular threshold, the probabilities are ordered \(0.02275<0.04743<0.06767\). This conclusion is about these specified models and this threshold; it is not a substitute for evaluating a new threshold under its own parameters.

Positive-Support Models and Their Thresholds

For lognormal, Weibull, and Pareto models, the threshold and the positive scale quantities must be expressed in compatible units. A ratio such as \(t/\lambda\) or \(x_m/t\) is meaningful only when its numerator and denominator refer to the same units. In a lognormal model, the ratio \(t/u\) makes the logarithm dimensionless.

Worked Example: Comparing Lognormal, Weibull, and Pareto Tails

Suppose the threshold is \(10\) cm. Let \(X_1\) be lognormal with \(\log(X_1/(1\text{ cm}))\sim N(\log 5,0.5^2)\); let \(X_2\) be Weibull with shape \(k=2\) and scale \(\lambda=8\) cm; and let \(X_3\) be Pareto Type I with minimum \(x_m=2\) cm and shape \(\alpha=2.5\). Find and compare the three exceedance probabilities.

Lognormal. Since the threshold and reference unit are both expressed in centimetres, the standardized log-threshold is

z=\frac{\log(10/1)-\log 5}{0.5} =\frac{\log 2}{0.5} \approx\frac{0.69315}{0.5} \approx1.3863.

Consequently,

P(X_1>10\text{ cm})=1-\Phi(1.3863)\approx 0.08283.

Weibull. The threshold-to-scale ratio is \(10/8=1.25\), so

P(X_2>10\text{ cm}) =\exp[-(10/8)^2] =\exp(-1.5625) \approx 0.20961.

Pareto. The threshold is in the support because \(10\text{ cm}\geq2\text{ cm}\). Its tail probability is

P(X_3>10\text{ cm}) =(2/10)^{2.5} =0.2^{2.5} =0.04\sqrt{0.2} \approx0.01789.

For these parameter choices, the Weibull probability is largest, followed by the lognormal probability and then the Pareto probability. Each value comes from a different formula; the ranking should not be inferred from the family names alone.

Unit Conversion in Lognormal Calculations

A lognormal model uses the logarithm of a positive quantity relative to a reference unit. When converting the recorded numerical values to a different unit, the reference unit must be converted as well. This keeps the standardized threshold—and hence the physical probability—unchanged.

Result: If \(X'=cX\) for \(c>0\), then \(P(X'>ct)=P(X>t)\). For a lognormal variable with reference unit \(u\), using the correspondingly converted reference unit \(u'=cu\) gives \(\log(X'/u')=\log(X/u)\). If instead the reference unit \(u\) is held fixed while the variable is multiplied by \(c\), then \(\log(X'/u)=\log(X/u)+\log c\), so the log-location shifts by \(\log c\).

The probability identity follows because \(c>0\), so \(X'>ct\) is exactly the event \(cX>ct\), which is equivalent to \(X>t\). For the lognormal statements, substitute \(X'=cX\) and \(u'=cu\) to obtain

\log(X'/u')=\log\!\left(\frac{cX}{cu}\right)=\log(X/u).

If the reference unit is fixed instead, direct substitution gives

\log(X'/u)=\log(cX/u)=\log(X/u)+\log c.

These are distinct operations: a consistent change of measurement unit changes both the numerical measurement and its reference unit; multiplying the quantity while keeping the reference unit fixed shifts the log-location. Confusing them creates a false change in a probability after a mere unit conversion.

Worked Example: The Same Lognormal Tail in Centimetres and Millimetres

Let \(X\) be a length satisfying \(\log(X/(1\text{ cm}))\sim N(\log 12,0.3^2)\). Find \(P(X>15\text{ cm})\), then verify the result when the length is expressed in millimetres.

In centimetres, the standardized threshold is

z_{\mathrm{cm}} =\frac{\log(15/1)-\log 12}{0.3} =\frac{\log(15/12)}{0.3} =\frac{\log(1.25)}{0.3} \approx\frac{0.22314}{0.3} \approx0.7438.

Thus \(P(X>15\text{ cm})=1-\Phi(0.7438)\approx0.2285\).

In millimetres, the numerical length is multiplied by \(10\), and the reference unit becomes \(1\) mm. The log-location is now \(\log 120=\log 12+\log 10\), because

\log(X/(1\text{ mm})) =\log(X/(1\text{ cm}))+\log 10.

The threshold is \(150\) mm. Its standardized value is therefore

z_{\mathrm{mm}} =\frac{\log 150-\log 120}{0.3} =\frac{\log(150/120)}{0.3} =\frac{\log(1.25)}{0.3} \approx0.7438.

The resulting tail probability is again approximately \(0.2285\), as it must be: the physical event “length greater than 15 cm” is the same event as “length greater than 150 mm.”

A Reliable Calculation Routine

A short sequence of checks reduces errors across all six families. First identify whether the event asks for an upper tail, lower tail, or interval. Then check support and units before substituting values. For the normal and lognormal families, calculate the standardized value explicitly; for the other families, preserve the parameterization shown in the model.

Write the event.
Translate the wording into notation, such as \(P(X>t)\) or \(P(X\leq t)\), before selecting a formula.

Check the model and support.
Identify the named family and its parameters. Check whether the threshold lies in the range where the desired formula applies.

Check units and substitute.
Match each threshold to its location, scale, or reference unit. Show the standardized value or ratio so that the substitution can be checked.

Evaluate and interpret.
Use the appropriate CDF or survival formula, round at the end, and state the probability in the context of the specified threshold.

Common Mistake: Mixing Up Tails or Units

A frequent error is to calculate \(F(t)\) when the question asks for an exceedance probability. For example, the normal calculation at \(z=2\) gives \(\Phi(2)\approx0.97725\), but this is \(P(Z\leq2)\), not \(P(Z>2)\). The requested upper tail is \(1-\Phi(2)\approx0.02275\). Another error is to insert a threshold in millimetres into a scale parameter stated in centimetres without converting one of them. A ratio of \(150/8\) is not the Weibull threshold-to-scale ratio if \(150\) is in millimetres and \(8\) is in centimetres; convert to \(15/8\) or \(150/80\) first.

For a lognormal calculation, it is also incorrect to say that the log-location necessarily gains \(\log c\) in a consistent unit conversion. That shift occurs when multiplying the variable by \(c\) while keeping the reference unit fixed. If the reference unit is multiplied by \(c\) as well, the ratio inside the logarithm is unchanged. State which operation is being performed before adjusting any lognormal parameter.

Key takeaway: A tail probability is determined by the event, the specified distribution, and correctly matched units. Write the event first, use the family’s proper CDF or survival formula, and verify every standardized difference or scale ratio before interpreting the result.

Check Your Understanding

For each question, identify the relevant formula and show the substitution before rounding.

If \(X\sim N(10,4^2)\), what is \(P(X>14)\), and what standardized threshold do you use?
A logistic variable has location \(3\) and scale \(2\). Write the formula for \(P(X>7)\) and evaluate it approximately.
A Weibull variable has shape \(1.5\) and scale \(6\) hours. What is the probability of exceeding \(9\) hours?
A Pareto Type I variable has minimum \(5\) cm and shape \(2\). What is \(P(X>10\text{ cm})\)?
When a lognormal length is converted from centimetres to millimetres, what happens to its log-location if the reference unit is converted consistently? What if the reference unit is held fixed?