Question

Find density function ,mean, variance of the distribution f(x)={1-e^-2x ,if x>0

Answer 1

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