An exponential random variable X depends on one parameter, which we here label L. In this case, E[X]=1/L and Var(X)=1/L^2.
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Expression 23: "L" equals 1L=1
negative 10−10
1010
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Expression 24: "f" left parenthesis, "x" , right parenthesis equals "L" exponent left parenthesis, negative "L" "x" , right parenthesis left brace, "x" greater than or equal to 0 , right brace Has graph. To audio trace, press ALT+T.fx=Lexp−Lxx≥0
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The sum of N independent exponential random variables is a gamma random variable.
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Expression 26: "f" Subscript, "s" , Baseline left parenthesis, "x" , right parenthesis equals StartFraction, "L" exp left parenthesis, negative "L" "x" , right parenthesis left parenthesis, "L" "x" , right parenthesis Superscript, "N" minus 1 , Baseline Over left parenthesis, "N" minus 1 , right parenthesis ! , EndFraction left brace, "x" greater than or equal to 0 , right brace Has graph. To audio trace, press ALT+T.fsx=Lexp−LxLxN−1N−1!x≥0
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Normalized distribution of sum
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Expression 28: "f" Subscript, "s" , Baseline left parenthesis, StartFraction, StartRoot, "N" , EndRoot Over "L" , EndFraction "x" plus StartFraction, "N" Over "L" , EndFraction , right parenthesis StartFraction, StartRoot, "N" , EndRoot Over "L" , EndFraction Has graph. To audio trace, press ALT+T.fsNLx+NLNL