Answer
Mean = μ = ∑X * P(X)
= (4*0.4)+(5*0.3)+(6*0.1)+(8*0.15)+(10*0.05)
= 5.4
Variance = $ σ ^{2}$ = ∑[$X ^{2}$ * P(X)] - $ μ ^{2}$
=[ $4 ^{2}$ *0.4 +$ 5 ^{2}$ * 0.3 + $6 ^{2}$ * 0.1 + $ 8 ^{2}$ *0.15 + $10 ^{2}$ *0.05 ] -$ 5.4 ^{2}$
=32.1-29.16
=2.94
Standard Deviation = σ = $ \sqrt σ ^{2}$ = $\sqrt 2.94$ = 1.7146
$P(X\geq 6)$ = P(X=6)+P(X=8)+P(X=10)
=0.1+0.15+0.05 = 0.25
Work Step by Step
Mean = μ = ∑X * P(X)
= (4*0.4)+(5*0.3)+(6*0.1)+(8*0.15)+(10*0.05)
= 5.4
Variance = $ σ ^{2}$ = ∑[$X ^{2}$ * P(X)] - $ μ ^{2}$
=[ $4 ^{2}$ *0.4 +$ 5 ^{2}$ * 0.3 + $6 ^{2}$ * 0.1 + $ 8 ^{2}$ *0.15 + $10 ^{2}$ *0.05 ] -$ 5.4 ^{2}$
=32.1-29.16
=2.94
Standard Deviation = σ = $ \sqrt σ ^{2}$ = $\sqrt 2.94$ = 1.71
Probability for one day
$P(X\geq 6)$ = P(X=6)+P(X=8)+P(X=10)
=0.1+0.15+0.05 = 0.30
Probability for 3 days in a row
(0.30)^3=0.027
