Life of batteries (Probability)

[ja404]

Homework Statement

The lifetime of batteries are independent random variables Xi, i= 1,2,... each having exponential distribution given by the density f(x) = 2e^-2x, x>0, 0 elsewhere.
If a flashlight needs two batteries to work, then the time that the flashlight can operate is a random variable Y=min(X1,X2). Find the CDF and PDF of Y

Homework Equations

Xi, i= 1,2,...
Xi ~ f(x) = 2e^-2x, x>0, 0 elsewhere
Y=min(X1,X2)

The Attempt at a Solution

What I'm not sure about is the initial approach to this problem. I'm not sure if I should treat it as an order statistics problem in which case i'd be trying to use the equation:

g(y) = n((1-Fx(y))^(n-1))f(y)

though I'm not sure if Fx(y) and f(y) are just the respective cdf and pdf of X.
I know if i have the pdf, I can get the cdf easily.

 

[Avodyne]

Do you know what a Dirac delta function is? Given a PDF f(x1,x2) for two variables x1 and x2, the PDF g(y) for any function y=h(x1,x2) of these is given by
$$g(y) = \int dx_1\,dx_2\,f(x_1,x_2)\delta(y-h(x_1,x_2))$$
where $\delta(y)$ is the Dirac delta function, and where x1 and x2 are integrated over their complete ranges (whatever they are).

 

[ja404]

Thanks for the help. I don't recall covering the dirac delta at all but I'm trying to use it and we'll see how it works. I was thinking there might be a transform involved as well? Thanks again, I appreciate it.

 

[Avodyne]

I was able to use this method to solve the problem, but it's tricky. There's probably (no pun intended) a method that avoids it; maybe someone else can help.

 

[EnumaElish]

Since Y is min(X1, X2), and if Prob{Xi < x) = F(x), what is the probability Prob{min(X1,X2) < y), in terms of the F function?

 

[ja404]

I just can't figure out how to deal with that "min". I know that if say Y=X1^2 then I would just follow:

cdf = Fy(y) = P(Y<y) = P(X^2<y) = P(X<Y^1/2) etc, but I don't know how to represent that 'minimum' when I'm trying to solve the inequality for X.

Thanks again.

 

[EnumaElish]

If the minimum of X1 and X2 is less than y, then it cannot be the case that "both X1 > y and X2 > y."

 

[ja404]

Would it correct to say then P(Y1>y) = P(X1>Y)*P(X2>Y) ?
Thanks.

 

[EnumaElish]

Exactly, or P(Y<y) = 1 - (1-P(X1<Y))*(1-P(X2<Y)).

 

[ja404]

Ok, I'm coming up with a pdf of:

(4e^-2y)*(1+e^-2y)

OR

(4e^-2y)+(4e^-4y)

Thoughts?

 

[Avodyne]

Not what I got. And I got the same answer with my fancy delta-function method and EmulaFish's simpler method.

Can you show your work?

 

[ja404]

well after talking to my professor, he said I should be using the order statistic technique

pdf of Y=min(X1,X2...Xn) = g(y) = n*[(1-Fx(y))^n-1]*fx(y)

where n=2 for this case
Fx(y) is cdf of X evaluated at y = -e^-2y
fx(y) is pdf of X evaluated at y = 2e^-2y

I put those into g(y) and that's how I came up with that.
Thanks again.

 

[ja404]

Any ideas on what I might be doing wrong? Thanks.

 

[ja404]

I figured out one thing I've been doing wrong. My new answer for pdf of Y=min(X1,X2) is

4e^-4y

I'm pretty confident about that one.

 

[Avodyne]

That's what I got!