Probability related to Normal Distribution

[songoku]

Homework Statement

Variable A has mean 55 and variance 9, variable B has mean 65 and variance 25. If A and B are normally distributed, find P (B > A)

Homework Equations

z = (x - μ) / σ

The Attempt at a Solution

Can this be solved? What is the meaning of P (B > A)? The probability of random variable B is bigger than random variable A?

Thanks

 

[hikaru1221]

I suppose the problem assumes A and B are statistically independent. Then the formal way to do this is:
$P(B>A) = \int_{b=-\infty}^{+\infty} \int_{a=-\infty}^{b}f_A(a)f_B(b)dadb$
This is a very nasty integral.

 

[Ray Vickson]

Homework Statement

Variable A has mean 55 and variance 9, variable B has mean 65 and variance 25. If A and B are normally distributed, find P (B > A)

Homework Equations

z = (x - μ) / σ

The Attempt at a Solution

Can this be solved? What is the meaning of P (B > A)? The probability of random variable B is bigger than random variable A?

Thanks

Yes, P(B > A) means exactly what you said. It can be solved quite easily if A and B are independent, but I cannot give more details now: first you have to make a serious attempt to solve it yourself.

 

[songoku]

I suppose the problem assumes A and B are statistically independent. Then the formal way to do this is:
$P(B>A) = \int_{b=-\infty}^{+\infty} \int_{a=-\infty}^{b}f_A(a)f_B(b)dadb$
This is a very nasty integral.

Sorry I don't get the idea how to do the integration and why can it be like that

Yes, P(B > A) means exactly what you said. It can be solved quite easily if A and B are independent, but I cannot give more details now: first you have to make a serious attempt to solve it yourself.

I don't have idea how to start. The question I saw until now always contained number such as P (B > 23). Please give me clue how to start solving this problem. What is the first step I need to take? I can't find the value of z because I don't know the value of the random variable.

Thanks

 

[Ray Vickson]

Sorry I don't get the idea how to do the integration and why can it be like that



I don't have idea how to start. The question I saw until now always contained number such as P (B > 23). Please give me clue how to start solving this problem. What is the first step I need to take? I can't find the value of z because I don't know the value of the random variable.

Thanks

Look at the random variable Y = B - A.

 

[songoku]

Look at the random variable Y = B - A.

OK maybe I get it

Let: Y = B - A

E(Y) = E(B) - E(A) = 10
σ(Y) = σ(B) + σ(A) = 34

P(B > A) = P(B - A > 0) = P(Y > 0) = P(Z > -1.715) = 0.9568

Do I get it right?

Thanks

 

[Ray Vickson]

OK maybe I get it

Let: Y = B - A

E(Y) = E(B) - E(A) = 10
σ(Y) = σ(B) + σ(A) = 34

P(B > A) = P(B - A > 0) = P(Y > 0) = P(Z > -1.715) = 0.9568

Do I get it right?

Thanks

It looks right.

 

[songoku]

It looks right.

ok thanks