Solved problems on probability and statistics

PROBLEM I
1) θ=0.2 and n=7
P ( X=3 )=7C3
( θ )3 (1−θ)4
P ( X=3 ) Where X ~ Binomial (n=7, θ=0.2 ¿
Answer) a)
2) P ( A )=0.6∧P ( B )=0.1
P ( A ∩B ) =0.08
P ( A ∪ B ) =P ( A ) + P ( B ) −P ( A ∩ B )
P ( A ∪ B )=0.62
Since,
P ( A ∩B ) ≠ P ( A ) × P ( B )
And,
P ( A ∪ B ) ≠ P ( A ) + P ( B )
So it is neither independent nor mutually exclusive.
Answer) e)
3) P ( X ≥2 )=1−P ( X =0 )−P ( X =1 )
P ( X ≥2 )=1−0.1−0.3=0.6
Answer) d)
4) f X ( x )=
{ d
dx
x
4 0 ≤ x <1
d
dx
x4 + 4
20 1 ≤ x< 2
d
dx 1 x ≥2
f X ( x )=
{ 1
4 0 ≤ x <1
x3
5 1 ≤ x <2
0 x ≥2
Answer) b)

5) 6
20 × 5
19 × 4
18
(6
3 )
(20
3 )
Answer) d)
6) P ( Good Risk ) =0.4
P ( Bad Risk ) =0.6
P ( NoCitation∨Good Risk ) =1−0.1=0.9
P ( NoCitation∨Bad Risk ) =1−0.5=0.5
P ( NoCitation ) =P ( NoCitation∨Good Risk ) ×
P ( Good Risk ) + P ( NoCitation∨Bad Risk ) × P ( Bad Risk ) = ( 0.4 ×0.9 ) + ( 0.6 ×0.5 ) =0.66
Answer) b)
7) f X (x )= { x
2 0 ≤ x <2
0 x ≥2
∫
0
x
x
2 dx=0.75
x2
4 =0.75
x= √4 ×0.75
Answer) c)
8) P ( X >Y ) = 4
24 + 2
24 + 1
24 = 7
24
Answer) a)
PROBLEM II