1 With a Swedish Kroner the likelihood of getting Heads when

1. With a Swedish Kroner, the likelihood of getting Heads when it is spun on edge is 0.2. If X is the random variable where X(H)=1, X(T) = -1, find the expected value E(X), the variance Var(X), and express X in its standard form.

2. a)In rolling a fair die, the probability of any number appearing is 1/6. Find the expected value and variance.

b) Consider two fair dice, the number 1 cannot occur, ao this event probability is 0. What\'s probability of 2, 3, 4.... Find the expected value and variance.

3. Show that Var(aX + b) = a^2*Var(X). The standard form is Z = (X-E(X))/ where ^2 = Var(X). Compute Var(Z).

Solution

X(H)=1 with pobability 0.2

X(T)=-1 with probability 1-0.2=0.8

E(X)=1(0.2)+(-1)(0.8)=-0.6

Variance(x)=E(X²)-[E(X)]²

                =1²(0.2)+(-1)²0.8-(0.6)²

                 =0.2+0.8-0.36=0.64

Since E(X) should be non negative, so we express X in standard form

Standard form of any random variable X is :

Y=(X-E(X))/(Variance(X) )^1/2

Y=X-(-0.6)/(0.64)^1/2

Y=(X+0.6)/0.8

E(Y) should be 0 and Variance (Y) should be 1 for this to be standard form

E(Y)=[E(X)+0.6]/0.8= -0.6+0.6/0.8=0

Variance(Y)=Variance(X+0.6)/0.8

                 =[Variance(x)+Variance(0.6])/0.8²

                     =[0.64+0]/0.64=1