Question

Problem 3 (Needed for Problem 4) A continuous random variable X is said to have an exponential distribution, written Exp(X), if its probability density function f is such that le- if > 0 10 if x < 0 f(0) = 0 where > 0 is a real number. 1. Compute the mean of X 2. Compute the variance of X 3. Compute the cumulative distribution function F of X. Use this to show that for any real numbers s and t such that s> 0 and t > 0, P(X > s +t|X >t) = e-s = P(X> s) This last property says that the random variable X is memoryless. Problem 4 (Needs results from Problem 3) One often models the lifetime of galaxies with exponential laws (see problem 3). Astronomers have found a galaxy for which they estimate that the probability of collapse within the next 1 million years is equal to 0.000002%. 1. Determine the parameter for the exponential law of the random variable X corresponding to the lifetime of the galaxy. 2. What is the expected lifetime of the galaxy? 3. What is the probability that the galaxy collapses within the next 3 billion years? 4. What is the probability that the galaxy is still present in 10 billion years?

Answer 1

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Ő FELCA) = P[X mo 3 = x 5 o te de -de 7 E-[ediyor -- [ 2*xo_i] -40 P[x> 8++\xxt] = P[x>stt n x>d] . :P[x >] i = P[x> St] P[x>t ] = [1- Fx Ce++)]. Ti - Fx () JOONPDAP, 3... -868++) - (+) = ěts CS Scanned PC Anne

(4) cet xi Time of collapse in a Million Mead 1) P[X3 1] = 0.0000000 2 l e 0.00000002 ē - 0.99999998. = 2.0000000 2 Xio D) E(X)= * = 49,999,949, 5 3) P[<< 3000] = inē - 0.999940002 = 5.999x105 1) P(x>10,000] - ötllere = 0.99986002 a Scanned with CamScanner