Can you explain the inequalities in exponential functions?

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The discussion focuses on understanding the inequalities involving exponential functions, specifically 1-exp(-μt) ≤ μt and (1-exp(-μt))exp(-λt) ≥ μt - (μ^2t^2/2)(1-λt). Participants seek clarification on why these inequalities are true, as they appear in lecture notes. The request emphasizes the need for a mathematical explanation rather than a general overview. A clear understanding of the properties of exponential functions and their behavior in these contexts is essential for grasping the inequalities. The conversation highlights the importance of foundational knowledge in calculus and exponential growth for interpreting these mathematical statements.
stukbv
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Hello, could somebody please explain to me how

1-exp(-μt) ≤ μt

and similarly

(1-exp(-μt))exp(-λt) ≥ μt-μ2t2\2)(1-λt)

Thanks a lot
 
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You haven't said what you want explained...
 
Sorry, I just want to know why they're true, they are in my lecture notes and I can't work out why we know these inequalities hold.
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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