Continues probability
Solution :
Given that ,
mean = = 20
variance = 9
standard deviation = = 9 = 3
X N( = 20 , = 3)
a)
P(X 10) = P((X - ) / (10 - 20) / 3)
= P(z -3.33)
Using standard normal table,
=0.0004
Probability = 0.0004
b)
P(15 X 20) = P((15 - 20)/ 3 (X - ) / (20 - 20) / 3 )
= P( -1.67 z 0)
= P(z 0) - P(z -1.67)
Using standard normal table,
= 0.5 - 0.0475
= 0.4525
Probability = 0.4525
c)
Given that ,
mean = = 2
variance = 12
standard deviation = =12 = 3.4641
Y N( = 2 , = 3.4641)
Z = 2X + Y + 3
E(Z) = E( 2X + Y + 3)
= E(2X) + E(Y) + 3 by additive property of expectation
=2E(X) + E(Y) + 3
= 2*20 + 2 + 3
= 40+5
= 45
E(Z) = 45
V(Z) = V(2X + Y + 3)
= V(2X) + V(Y) +0 , by additive property of variance.
Since X and Y are independent to each other.
= 4V(X) + 12
=4*9 + 12
=36+12
= 48
V(Z) = 48
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