Sharp inequalities of Bienaymé–Chebyshev and Gauß type for possibly asymmetric intervals around the mean

Sharp upper bounds are proved for the probability that a standardized random variable takes on a value outside a possibly asymmetric interval around 0. Six classes of distributions for the random variable are considered, namely the general class of ‘distributions’, the class of ‘symmetric distributions’, of ‘concave distributions’, of ‘unimodal distributions’, of ‘unimodal distributions with coinciding mode and mean’, and of ‘symmetric unimodal distributions’. In this way, results by Gauß (Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores 5:1–58, 1823), Bienaymé (C R Hebd Séance Acad Sci Paris 37:309–24, 1853), Bienaymé (C R Hebd Séance Acad Sci Paris 37:309–24, 1853), Chebyshev (Journal de mathématiques pures et appliqués (2) 12:177–184, 1867), and Cantelli (Atti del Congresso Internazionale dei Matematici 6:47–59, 1928) are generalized. For some of the known inequalities, such as the Gauß inequality, an alternative proof is given

Burden and risk factors for Pseudomonas aeruginosa community-acquired pneumonia:a Multinational Point Prevalence Study of Hospitalised Patients

Pseudornonas aeruginosa is a challenging bacterium to treat due to its intrinsic resistance to the antibiotics used most frequently in patients with community-acquired pneumonia (CAP). Data about the global burden and risk factors associated with P. aeruginosa-CAP are limited. We assessed the multinational burden and specific risk factors associated with P. aeruginosa-CAP. We enrolled 3193 patients in 54 countries with confirmed diagnosis of CAP who underwent microbiological testing at admission. Prevalence was calculated according to the identification of P. aeruginosa. Logistic regression analysis was used to identify risk factors for antibiotic-susceptible and antibiotic-resistant P. aeruginosa-CAP. The prevalence of P. aeruginosa and antibiotic-resistant P. aeruginosa-CAP was 4.2% and 2.0%, respectively. The rate of P. aeruginosa CAP in patients with prior infection/colonisation due to P. aeruginosa and at least one of the three independently associated chronic lung diseases (i.e. tracheostomy, bronchiectasis and/or very severe chronic obstructive pulmonary disease) was 67%. In contrast, the rate of P. aeruginosa-CAP was 2% in patients without prior P. aeruginosa infection/colonisation and none of the selected chronic lung diseases. The multinational prevalence of P. aeruginosa-CAP is low. The risk factors identified in this study may guide healthcare professionals in deciding empirical antibiotic coverage for CAP patients