How to find probability density function, variance of distribution F(X)=1-e^-2x if X>0
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Answer:
f(x)={
αe
−αx
ifx>0
0otherwise
E(X)=∫
−∞
∞
xf(x)dx=∫
0
∞
x(α)(e
−αx
)dx
=α∫
0
∞
xe
−αx
dx=
α
2
α(1)
=
α
1
[By Bernoulli's formula ∫
0
∞
x
n
e
−αx
dx=
a
n+1
n
]
E(X
2
)=∫
0
∞
x
2
(αe
−αx
)dx
[∫
0
∞
x
n
e
−αx
dx=
a
n+1
n
]=α∫
0
∞
x
2
e
−αx
dx=
α
3
α(2)
=
α
2
2
[By Bernoulli's formula]
Mean=E(X)=
α
1
Variance =E(X
2
)−[E(X)]
2
=
α
2
2
−
α
2
1
=
α
2
1
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