Random variable with exponential distribution.
Let X
be random variable with exponential distribution E(2)
and let Y
be another random variable such that
Y=max(X2,
X+1
2
)
Find the distribution for random variable Y
.
Distribution for X
is fX(x)=2e−2x,x>0
and zero otherwise.
Now, for variable Y
we have that it's distribution is zero whenever y≤
1
4
For y=t>
1
2
we have the following:
FY(t)=∫
2t−1
0
fX(x)dx=1−e2−4t
Similarly, for y=t>1
we have
FY(t)=∫
√
t
0
fX(x)dx=1−e−2
√
t
But, i cannot understand what happens in case that y
takes random value on interval (
1
4
,
1
2
)
. It's the black line on the graph. How can i handle situations like this? Any help appreciated!
be random variable with exponential distribution E(2)
and let Y
be another random variable such that
Y=max(X2,
X+1
2
)
Find the distribution for random variable Y
.
Distribution for X
is fX(x)=2e−2x,x>0
and zero otherwise.
Now, for variable Y
we have that it's distribution is zero whenever y≤
1
4
For y=t>
1
2
we have the following:
FY(t)=∫
2t−1
0
fX(x)dx=1−e2−4t
Similarly, for y=t>1
we have
FY(t)=∫
√
t
0
fX(x)dx=1−e−2
√
t
But, i cannot understand what happens in case that y
takes random value on interval (
1
4
,
1
2
)
. It's the black line on the graph. How can i handle situations like this? Any help appreciated!
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