The standard deviation for a random variable with probability density function f and mean µ is defined . Find the standard deviation for an exponential density function with mean .

Question

The standard deviation for a random variable with probability density function f and mean µ is defined  . Find the standard deviation for an exponential density function with mean  .

Quizplus · Accepted Answer

For an exponential density function with mean µ, the probability density function is f(x) = (1/µ) * e^(-x/µ). 
The standard deviation can be found using the formula: 
σ = sqrt(E[(X-µ)^2]) = sqrt(E[X^2 - 2Xµ + µ^2]) = sqrt(E[X^2] - 2µ^2 + µ^2) = sqrt(E[X^2] - µ^2)

To find E[X^2], we can integrate (x^2)(1/µ)e^(-x/µ) from 0 to infinity:

E[X^2] = ∫(0 to ∞) (x^2)(1/µ)e^(-x/µ) dx  
= (2µ^2)

Therefore,

σ = sqrt(E[X^2] - µ^2) 
= sqrt(2µ^2 - µ^2) 
= sqrt(µ^2) 
= µ

Substituting µ = 11, we get σ = 11. Hence, the correct choice is C.

The Standard Deviation for a Random Variable with Probability Density