Answer:
E(X) = 1.9375
SD(X) = 1.20
Explanation:
Let :
G = girl ; B = Boy
P(G) = 0.5 ; P(B) = 0.5
Sample space :
G, BG, BBG, BBBG, BBBBG, BBBBB
Using the multiplication rule of independence :
P(G) = 0.5
P(BG) = 0.5 * 0.5 = 0.25
P(BBG) = 0.5 * 0.5 * 0.5 = 0.125
P(BBBG) = 0.5 * 0.5 * 0.5 * 0.5 = 0.0625
P(BBBBG) = 0.5 * 0.5 * 0.5 * 0.5 * 0.5 = 0.03125
P(BBBBB) = 0.5 * 0.5 * 0.5 * 0.5 * 0.5 = 0.03125
Creating a probability distribution table :
X = 1, 2, 3, 4, 5
For X = 5 ;
P(BBBBG) + P(BBBBB) = 0.03125 + 0.03125 = 0.0625
X :___ 1 ____ 2 ____ 3 _____ 4 _____ 5
P(X) : 0.5 __ 0.25 _ 0.125 _0.0625 _0.0625
The expected value E(X) :
√(ΣX * P(X)) = (1*0.5)+(2*0.25)+(3*0.125)+(4*0.0625)+(5*0.0625) = 1.9375
The standard deviation SD(x) :
√(ΣX²*p(x) - E(X)²
√((1^2*0.5)+(2^2*0.25)+(3^2*0.125)+(4^2*0.0625)+(5^2*0.0625) - 1.9375^2)
√(5.1875 - 3.75390625)
√1.43359375
SD(X) = 1.197
SD(X) = 1.20