In March 1995 the space shuttle carried an experiment designed by a Harvard student who studies the growth of crystals.Suppose the probability density function of the length, x cm, of a crystal grown in space is modeled by $p(x)=x e^{-x}$ for $x \geq 0$ .The cumulative distribution function giving the probability that a crystal has length$\le$ t cm is represented by P(t).Which of the following formulas gives the cumulative distribution function P(t)for t $\ge$ 0?
A) $\frac{t-1}{e^{t}}$
B) $-\frac{t+1}{e^{t}}$
C) $1+\frac{t-1}{e^{t}}$
D) $1-\frac{t+1}{e^{t}}$

Question

Quizplus · Accepted Answer

The cumulative distribution function $P(t)$ is found by integrating the probability density function $p(x) = x e^{-x}$ from 0 to $t$. The integral of $x e^{-x}$ is $-(x+1)e^{-x} + C$. Evaluating this from 0 to $t$ gives $P(t) = [-(t+1)e^{-t} + C] - [-(0+1)e^{0} + C] = 1 - (t+1)e^{-t}$, which matches option D.

In March 1995 the Space Shuttle Carried an Experiment Designed p(x)=xe−xp(x)=x e^{-x}p(x)=xe−x

In March 1995 the Space Shuttle Carried an Experiment Designed $p(x)=x e^{-x}$