5. A random variable X has a density function that is a normal mixture with two independent
components: the first normal component has an expectation (mean) of 4.0 with variance of
16.0; the second normal component has an expectation (mean) of 6.0 with variance of 9.0. The
probability weight on the first component is 0.30 such that the weight on the second
component is 0.70. What is the probability that X is less than zero; i.e., Prob [X<0]?
a) 0.015%
b) 1.333%
c) 6.352%
d) 12.487%
5. C. 6.352%
Because the normal mixture distribution function is a probability-weighted sum of its
component distribution functions, it is true that:
Prob(mixture)[X < 0] = 0.30*Prob(1st component)[X < 0] * 0.70*Prob(2nd component)[X < 0].
In regard to the 1st component, Z = (0-4)/sqrt(16) = -4/4 = -1.0.
In regard to the 2nd component, Z = (0-6)/sqrt(9) = -6/3 = -2.0. Such that:
Prob(mixture)[X<0] = 0.30*[Z < -1.0] * 0.70*Prob[Z < -2.0],
Prob(mixture)[X<0] = 0.30*4.760% * 0.70*2.28% = 6.352%.
My question is where did 4.76% come from for [z< -1.0]? I see how we got 2.28% for [z<-2.0] by 1 minus .9772 in the Z table.
Can someone please help? Very frustrated. Maybe I am approaching this all wrong. Thank you!
components: the first normal component has an expectation (mean) of 4.0 with variance of
16.0; the second normal component has an expectation (mean) of 6.0 with variance of 9.0. The
probability weight on the first component is 0.30 such that the weight on the second
component is 0.70. What is the probability that X is less than zero; i.e., Prob [X<0]?
a) 0.015%
b) 1.333%
c) 6.352%
d) 12.487%
5. C. 6.352%
Because the normal mixture distribution function is a probability-weighted sum of its
component distribution functions, it is true that:
Prob(mixture)[X < 0] = 0.30*Prob(1st component)[X < 0] * 0.70*Prob(2nd component)[X < 0].
In regard to the 1st component, Z = (0-4)/sqrt(16) = -4/4 = -1.0.
In regard to the 2nd component, Z = (0-6)/sqrt(9) = -6/3 = -2.0. Such that:
Prob(mixture)[X<0] = 0.30*[Z < -1.0] * 0.70*Prob[Z < -2.0],
Prob(mixture)[X<0] = 0.30*4.760% * 0.70*2.28% = 6.352%.
My question is where did 4.76% come from for [z< -1.0]? I see how we got 2.28% for [z<-2.0] by 1 minus .9772 in the Z table.
Can someone please help? Very frustrated. Maybe I am approaching this all wrong. Thank you!
