probability distribution and distribution function
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Given that $X$ is a random variable with cdf
$F(x)= 1-e^{-x}$ when $x>0$ and $0$ when $x le 0$, find $P(0 le e^X le 4)$.
I know the formula for $P(a le X le b)$ but here how I convert $e^X$ into $X$ . I assume that $y=e^X$ but in that case if I take log on both sides then it is not defined at $X=0$.
probability probability-distributions
edited Nov 22 at 18:26
Daniel
1,516210
asked Nov 22 at 17:56
tiger
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up vote
0
down vote
favorite
Given that $X$ is a random variable with cdf
$F(x)= 1-e^{-x}$ when $x>0$ and $0$ when $x le 0$, find $P(0 le e^X le 4)$.
I know the formula for $P(a le X le b)$ but here how I convert $e^X$ into $X$ . I assume that $y=e^X$ but in that case if I take log on both sides then it is not defined at $X=0$.
probability probability-distributions
edited Nov 22 at 18:26
Daniel
1,516210
asked Nov 22 at 17:56
tiger
13
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given that $X$ is a random variable with cdf
$F(x)= 1-e^{-x}$ when $x>0$ and $0$ when $x le 0$, find $P(0 le e^X le 4)$.
I know the formula for $P(a le X le b)$ but here how I convert $e^X$ into $X$ . I assume that $y=e^X$ but in that case if I take log on both sides then it is not defined at $X=0$.
probability probability-distributions
edited Nov 22 at 18:26
Daniel
1,516210
asked Nov 22 at 17:56
tiger
13
Given that $X$ is a random variable with cdf
$F(x)= 1-e^{-x}$ when $x>0$ and $0$ when $x le 0$, find $P(0 le e^X le 4)$.
I know the formula for $P(a le X le b)$ but here how I convert $e^X$ into $X$ . I assume that $y=e^X$ but in that case if I take log on both sides then it is not defined at $X=0$.
probability probability-distributions
probability probability-distributions
edited Nov 22 at 18:26
Daniel
1,516210
asked Nov 22 at 17:56
tiger
13
edited Nov 22 at 18:26
Daniel
1,516210
asked Nov 22 at 17:56
tiger
13
edited Nov 22 at 18:26
Daniel
1,516210
edited Nov 22 at 18:26
Daniel
1,516210
edited Nov 22 at 18:26
Daniel
1,516210
1,516210
asked Nov 22 at 17:56
tiger
13
asked Nov 22 at 17:56
tiger
13
asked Nov 22 at 17:56
tiger
13
13
add a comment |
add a comment |
1 Answer
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Since $exp(x)>0$ for any $x in mathbb{R}$, we have
$$P(0 le e^X le 4) = P(0<e^X le 4).$$
Moreover, we know that $log$ maps $(0,4]$ injective onto $(-infty,log(4)]$, because this function is monotonically increasing and continuous. Thus
$$P(0<e^X le 4) = P(X le log(4)) = F(log(4)) = 1-e^{-log(4)}=1-1/4=3/4.$$
answered Nov 22 at 18:03
p4sch
4,800217
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Since $exp(x)>0$ for any $x in mathbb{R}$, we have
$$P(0 le e^X le 4) = P(0<e^X le 4).$$
Moreover, we know that $log$ maps $(0,4]$ injective onto $(-infty,log(4)]$, because this function is monotonically increasing and continuous. Thus
$$P(0<e^X le 4) = P(X le log(4)) = F(log(4)) = 1-e^{-log(4)}=1-1/4=3/4.$$
answered Nov 22 at 18:03
p4sch
4,800217
add a comment |
up vote
1
down vote
Since $exp(x)>0$ for any $x in mathbb{R}$, we have
$$P(0 le e^X le 4) = P(0<e^X le 4).$$
Moreover, we know that $log$ maps $(0,4]$ injective onto $(-infty,log(4)]$, because this function is monotonically increasing and continuous. Thus
$$P(0<e^X le 4) = P(X le log(4)) = F(log(4)) = 1-e^{-log(4)}=1-1/4=3/4.$$
answered Nov 22 at 18:03
p4sch
4,800217
add a comment |
up vote
1
down vote
up vote
1
down vote
Since $exp(x)>0$ for any $x in mathbb{R}$, we have
$$P(0 le e^X le 4) = P(0<e^X le 4).$$
Moreover, we know that $log$ maps $(0,4]$ injective onto $(-infty,log(4)]$, because this function is monotonically increasing and continuous. Thus
$$P(0<e^X le 4) = P(X le log(4)) = F(log(4)) = 1-e^{-log(4)}=1-1/4=3/4.$$
answered Nov 22 at 18:03
p4sch
4,800217
Since $exp(x)>0$ for any $x in mathbb{R}$, we have
$$P(0 le e^X le 4) = P(0<e^X le 4).$$
Moreover, we know that $log$ maps $(0,4]$ injective onto $(-infty,log(4)]$, because this function is monotonically increasing and continuous. Thus
$$P(0<e^X le 4) = P(X le log(4)) = F(log(4)) = 1-e^{-log(4)}=1-1/4=3/4.$$
answered Nov 22 at 18:03
p4sch
4,800217
answered Nov 22 at 18:03
p4sch
4,800217
answered Nov 22 at 18:03
p4sch
4,800217
answered Nov 22 at 18:03
p4sch
4,800217
4,800217
add a comment |
add a comment |
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