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Let $X$ and $Y$ be two exponentially distributed and independent random variables with mean $α$ and $β$, respectively. If $Z$ = min $(X, Y)$, then the mean of $Z$ is given by

  1. $\left(\dfrac{1}{\alpha + \beta}\right)$
  2. $\min (\alpha, \beta)$
  3. $\left(\dfrac{\alpha\beta}{\alpha + \beta}\right)$
  4. $\alpha + \beta$
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4 Comments

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derivation is not in syllabus
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1 Answer

38 votes
38 votes
Best answer

Answer is (C)

$X$ is an exponential random variable of parameter λ when its probability distribution function is

$$f(x) = \begin{cases}\lambda e^{-\lambda x} & x \geq 0 \\ 0 & x < 0  \end{cases}$$

For a > 0, we have the cumulative distribution function

$$F_x(a) = \int_0^a f(x) dx = \int_0^a  \lambda e^{-\lambda x} dx = -e^{-\lambda x} \mid_0^a = 1 - e^ {-\lambda a}$$

So, 

$$P\left\{X < a \right \} = 1 - e^ {-\lambda a} $$ and 

$$P\left\{X > a \right \} = e^ {-\lambda a} $$

Now, we use $P \left \{X > a \right \}$ for our problem because our concerned variable $Z$ is min of $X$ and $Y$. 

For exponential distribution with parameter $\lambda$, mean is given by $\frac{1}{\lambda}$.
We have,

$P \left \{X > a \right \} = e^ {-\frac{1}{\alpha} a} $

$P \left \{Y > a \right \} = e^ {-\frac{1}{\beta} a} $

So, $\begin{align*}P\left \{Z > a \right \} &= P \left \{X > a \right \}  P \left \{Y > a \right \} \left(\because \text{X and Y are independent events and } \\Z > \min \left(X, Y \right) \right)\\&=e^ {-\frac{1}{\alpha} a}  e^ {-\frac{1}{\beta} a} \\&=e^{-\left(\frac{1}{\alpha} + \frac{1}{\beta} \right)a} \\&=e^{-\left(\frac{\alpha + \beta} {\alpha \beta} \right)a}\end{align*}$

This shows that $Z$ is also exponentially distributed with parameter $\frac{\alpha + \beta} {\alpha \beta}$ and mean $\frac{\alpha  \beta} {\alpha + \beta}$.

Ref: http://ocw.mit.edu/courses/mathematics/18-440-probability-and-random-variables-spring-2011/lecture-notes/MIT18_440S11_Lecture20.pdf 

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4 Comments

Such a simple a beautiful approach Arjun sir
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Thanks for sharing the PDF ayush, I didnt know about the memoryless property.
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I had some problem in understanding $Z>a$ iff $X>a$ and $Y>a$

So I tried to draw analogy using calculus:

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