Normalised total distribution on 156 sites looked as follows: 0%, 0, 0, 0, 0, 0.004506738, 0.009013475, 0.013520213, 0.01802695, 0.022533688, 0.031547163, 0.040560638, 0.049574113, 0.058587588, 0.067601064, 0.081121276, 0.094641489, 0.108161702, 0.126188652, 0.144215602, 0.162242553, 0.180269503, 0.202803191, 0.225336879, 0.247870566, 0.283924467, 0.319978368, 0.360539006, 0.401099644, 0.441660282, 0.48222092, 0.527288296, 0.576862409, 0.63094326, 0.685024111, 0.743611699, 0.802199288, 0.860786876, 0.919374465, 0.977962053, 1.041056379, 1.11316418, 1.203298932, 1.293433683, 1.383568435, 1.482716661, 1.599891838, 1.717067015, 1.83874893, 1.960430844, 2.100139709, 2.248862049, 2.402091126, 2.555320204, 2.713056019, 2.875298571, 3.042047862, 3.208797152, 3.384559917, 3.560322682, 3.740592185, 3.925368426, 4.128171617, 4.330974807, 4.542791473, 4.759114877, 4.984451755, 5.209788634, 5.43963225, 5.669475866, 5.90382622, 6.138176574, 6.38604714, 6.633917707, 6.881788273, 7.134165578, 7.391049619, 7.656947136, 7.945378341, 8.238316283, 8.535760963, 8.842219118, 9.157690748, 9.477669115, 9.806660958, 10.14466628, 10.48717833, 10.87025103, 11.25783046, 11.69047726, 12.14115102, 12.60534499, 13.07855244, 13.55626662, 14.0339808, 14.51620172, 15.03447654, 15.5572581, 16.0845464, 16.61183469, 17.14362973, 17.70697192, 18.30186128, 18.90125738, 19.50065348, 20.10455631, 20.75352652, 21.40700347, 22.10554779, 22.82211907, 23.57023751, 24.32736942, 25.0935148, 25.85966019, 26.68889991, 27.54968678, 28.47807472, 29.42448961, 30.39343819, 31.49308216, 32.59723286, 33.73293073, 34.8686286, 36.01784668, 37.17607824, 38.38839064, 39.63225021, 40.9617378, 42.32727928, 43.76042183, 45.25215197, 46.78444274, 48.33025373, 49.92113209, 51.67875975, 53.44540087, 55.25710938, 57.07332462, 58.98418135, 60.92207851, 62.96813737, 65.09982424, 67.2900987, 69.4848799, 71.68416783, 74.07724548, 76.52440398, 79.04817702, 81.72067241, 84.39767452, 87.11523728, 89.99954933, 93.08666456, 96.50727838, 100% (click here to see this data visualised to a chart)
Normalised total distribution on 156 sites looked as follows: 0%, 0, 0, 0, 0, 0.004506738, 0.009013475, 0.013520213, 0.01802695, 0.022533688, 0.031547163, 0.040560638, 0.049574113, 0.058587588, 0.067601064, 0.081121276, 0.094641489, 0.108161702, 0.126188652, 0.144215602, 0.162242553, 0.180269503, 0.202803191, 0.225336879, 0.247870566, 0.283924467, 0.319978368, 0.360539006, 0.401099644, 0.441660282, 0.48222092, 0.527288296, 0.576862409, 0.63094326, 0.685024111, 0.743611699, 0.802199288, 0.860786876, 0.919374465, 0.977962053, 1.041056379, 1.11316418, 1.203298932, 1.293433683, 1.383568435, 1.482716661, 1.599891838, 1.717067015, 1.83874893, 1.960430844, 2.100139709, 2.248862049, 2.402091126, 2.555320204, 2.713056019, 2.875298571, 3.042047862, 3.208797152, 3.384559917, 3.560322682, 3.740592185, 3.925368426, 4.128171617, 4.330974807, 4.542791473, 4.759114877, 4.984451755, 5.209788634, 5.43963225, 5.669475866, 5.90382622, 6.138176574, 6.38604714, 6.633917707, 6.881788273, 7.134165578, 7.391049619, 7.656947136, 7.945378341, 8.238316283, 8.535760963, 8.842219118, 9.157690748, 9.477669115, 9.806660958, 10.14466628, 10.48717833, 10.87025103, 11.25783046, 11.69047726, 12.14115102, 12.60534499, 13.07855244, 13.55626662, 14.0339808, 14.51620172, 15.03447654, 15.5572581, 16.0845464, 16.61183469, 17.14362973, 17.70697192, 18.30186128, 18.90125738, 19.50065348, 20.10455631, 20.75352652, 21.40700347, 22.10554779, 22.82211907, 23.57023751, 24.32736942, 25.0935148, 25.85966019, 26.68889991, 27.54968678, 28.47807472, 29.42448961, 30.39343819, 31.49308216, 32.59723286, 33.73293073, 34.8686286, 36.01784668, 37.17607824, 38.38839064, 39.63225021, 40.9617378, 42.32727928, 43.76042183, 45.25215197, 46.78444274, 48.33025373, 49.92113209, 51.67875975, 53.44540087, 55.25710938, 57.07332462, 58.98418135, 60.92207851, 62.96813737, 65.09982424, 67.2900987, 69.4848799, 71.68416783, 74.07724548, 76.52440398, 79.04817702, 81.72067241, 84.39767452, 87.11523728, 89.99954933, 93.08666456, 96.50727838, 100%
Normalised total distribution on 156 sites looked as follows: 0%, 0, 0, 0, 0, 0.004506738, 0.009013475, 0.013520213, 0.01802695, 0.022533688, 0.031547163, 0.040560638, 0.049574113, 0.058587588, 0.067601064, 0.081121276, 0.094641489, 0.108161702, 0.126188652, 0.144215602, 0.162242553, 0.180269503, 0.202803191, 0.225336879, 0.247870566, 0.283924467, 0.319978368, 0.360539006, 0.401099644, 0.441660282, 0.48222092, 0.527288296, 0.576862409, 0.63094326, 0.685024111, 0.743611699, 0.802199288, 0.860786876, 0.919374465, 0.977962053, 1.041056379, 1.11316418, 1.203298932, 1.293433683, 1.383568435, 1.482716661, 1.599891838, 1.717067015, 1.83874893, 1.960430844, 2.100139709, 2.248862049, 2.402091126, 2.555320204, 2.713056019, 2.875298571, 3.042047862, 3.208797152, 3.384559917, 3.560322682, 3.740592185, 3.925368426, 4.128171617, 4.330974807, 4.542791473, 4.759114877, 4.984451755, 5.209788634, 5.43963225, 5.669475866, 5.90382622, 6.138176574, 6.38604714, 6.633917707, 6.881788273, 7.134165578, 7.391049619, 7.656947136, 7.945378341, 8.238316283, 8.535760963, 8.842219118, 9.157690748, 9.477669115, 9.806660958, 10.14466628, 10.48717833, 10.87025103, 11.25783046, 11.69047726, 12.14115102, 12.60534499, 13.07855244, 13.55626662, 14.0339808, 14.51620172, 15.03447654, 15.5572581, 16.0845464, 16.61183469, 17.14362973, 17.70697192, 18.30186128, 18.90125738, 19.50065348, 20.10455631, 20.75352652, 21.40700347, 22.10554779, 22.82211907, 23.57023751, 24.32736942, 25.0935148, 25.85966019, 26.68889991, 27.54968678, 28.47807472, 29.42448961, 30.39343819, 31.49308216, 32.59723286, 33.73293073, 34.8686286, 36.01784668, 37.17607824, 38.38839064, 39.63225021, 40.9617378, 42.32727928, 43.76042183, 45.25215197, 46.78444274, 48.33025373, 49.92113209, 51.67875975, 53.44540087, 55.25710938, 57.07332462, 58.98418135, 60.92207851, 62.96813737, 65.09982424, 67.2900987, 69.4848799, 71.68416783, 74.07724548, 76.52440398, 79.04817702, 81.72067241, 84.39767452, 87.11523728, 89.99954933, 93.08666456, 96.50727838, 100% (click here to see this data visualised to a chart)
Brief overview for those having no time to drill into details of data provided in top answer.
Data was collected for 8 months total between August 2019 and March 2020, inclusive.
There were 11 eligible sites that had no hot questions at all in this period: beer, coffee, conlang, ebooks, eosio, expressionengine, freelancing, languagelearning, tor, vegan, windowsphone. 30 more sites had less than one hot question a month average. 26 more sites had less than one hot question a week average.
On the opposite angle, 99 sites had one or more hot questions a week average. Of these, 46 sites had one or more hot question a day: chemistry, bicycles, gaming, superuser, or, boardgames, judaism, apple, retrocomputing, ell, dba, english, money, chess, space, codegolf, blender, askubuntu, japanese, gis, stats, salesforce, aviation, music, politics, workplace, diy, academia, codereview, mathoverflow, unix, travel, physics, stackoverflow, electronics, scifi, worldbuilding, mathematica, puzzling, rpg, tex, math.
In order to get some kind of aggregate data about less represented sites, I first reordered the data by amount of hot questions from smallest to largest, then calculated the rolling total. Next I scaled these absolute numbers to a range of 0 to 100% (since I wanted relative comparison).
Also, after observing huge variation in numbers I decided to cut off and ignore six top and six bottom sites assuming that these might be outliers. This took away quite a lot of original data (top six sites contributed about 25% of all hot questions, go figure), but remaining amount still looked large enough to build a meaningful picture.
Normalised total distribution on 156 sites looked as follows: 0%, 0, 0, 0, 0, 0.004506738, 0.009013475, 0.013520213, 0.01802695, 0.022533688, 0.031547163, 0.040560638, 0.049574113, 0.058587588, 0.067601064, 0.081121276, 0.094641489, 0.108161702, 0.126188652, 0.144215602, 0.162242553, 0.180269503, 0.202803191, 0.225336879, 0.247870566, 0.283924467, 0.319978368, 0.360539006, 0.401099644, 0.441660282, 0.48222092, 0.527288296, 0.576862409, 0.63094326, 0.685024111, 0.743611699, 0.802199288, 0.860786876, 0.919374465, 0.977962053, 1.041056379, 1.11316418, 1.203298932, 1.293433683, 1.383568435, 1.482716661, 1.599891838, 1.717067015, 1.83874893, 1.960430844, 2.100139709, 2.248862049, 2.402091126, 2.555320204, 2.713056019, 2.875298571, 3.042047862, 3.208797152, 3.384559917, 3.560322682, 3.740592185, 3.925368426, 4.128171617, 4.330974807, 4.542791473, 4.759114877, 4.984451755, 5.209788634, 5.43963225, 5.669475866, 5.90382622, 6.138176574, 6.38604714, 6.633917707, 6.881788273, 7.134165578, 7.391049619, 7.656947136, 7.945378341, 8.238316283, 8.535760963, 8.842219118, 9.157690748, 9.477669115, 9.806660958, 10.14466628, 10.48717833, 10.87025103, 11.25783046, 11.69047726, 12.14115102, 12.60534499, 13.07855244, 13.55626662, 14.0339808, 14.51620172, 15.03447654, 15.5572581, 16.0845464, 16.61183469, 17.14362973, 17.70697192, 18.30186128, 18.90125738, 19.50065348, 20.10455631, 20.75352652, 21.40700347, 22.10554779, 22.82211907, 23.57023751, 24.32736942, 25.0935148, 25.85966019, 26.68889991, 27.54968678, 28.47807472, 29.42448961, 30.39343819, 31.49308216, 32.59723286, 33.73293073, 34.8686286, 36.01784668, 37.17607824, 38.38839064, 39.63225021, 40.9617378, 42.32727928, 43.76042183, 45.25215197, 46.78444274, 48.33025373, 49.92113209, 51.67875975, 53.44540087, 55.25710938, 57.07332462, 58.98418135, 60.92207851, 62.96813737, 65.09982424, 67.2900987, 69.4848799, 71.68416783, 74.07724548, 76.52440398, 79.04817702, 81.72067241, 84.39767452, 87.11523728, 89.99954933, 93.08666456, 96.50727838, 100%
From above, we can see that 86 least represented sites of 156 total contributed less than 10% hot questions, 106 contributed less than 20% questions and 135 - less than 50% questions.
21 top sites of 156 contributed over 50% hot questions. (Please keep in mind that this ignores top six and bottom six sites, otherwise the percentages for least exposed sites would look worse than here - as I mentioned above, ignored top 6 sites had about 25% of total questions.)