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Comparing random variables with a normal distribution

1. Homework Statement

You have 7 apples whose weight (in gram) is independent of each other and normally distributed, N([itex]\mu[/itex]= 150, [itex]\sigma[/itex]2 = 202).
You also have a cabbage whose weight is independent of the apples and N(1000, 502)

What is the probability that the seven apples will weigh more than the cabbage?

2. Homework Equations

Let X represent the weight of the seven apples combined, and Y the weight of the cabbage.
X~N(1050, 2800)
Y ~N(1000, 502)

3. The Attempt at a Solution

I have an easy time calculating the probability that a random variable will yield a number within a specific interval. For example I know how to get the probability that the 7 apples will weigh more that 1000g,
p(X > 1000) =
[itex]\varphi[/itex](X > (1000 - [itex]\mu[/itex]X)/[itex]\sigma[/itex]x) =
[itex]\Phi[/itex](0.94) = 0.8264, which I got from a chart for [itex]\Phi[/itex](x).

I am completely lost however on how to calculate the probability that a certain random variable will yield a bigger number that another random variable, both normally distributed but with different parameters: p(X > Y).

Thank you.
1. Homework Statement



2. Homework Equations



3. The Attempt at a Solution
 
21,992
3,272
Let [itex]X_i[/itex] be the distribution of the apples. And let Y be the distribution of the cabbage.

You know that [itex]X_i\sim N(150,20^2)[/itex] and [itex]Y\sim N(1000,50^2)[/itex].

Can you find out the distribution of -Y??
Can you use this to find out the distribution of [itex]X:=X_1+X_2+X_3+X_4+X_5+X_6+X_7-Y[/itex]??
Can you use this to find [itex]P(X\geq 0)[/itex]??
 
Thank you :)
I like how stuff is obvious when someone tells you.

-y ~ N(-1000, 502)

[itex]X[/itex]i ~ N(1050 - 1000, 2800 + 2500)

P( [itex]X[/itex][itex]\geq[/itex]0) = [itex]\Phi[/itex]([itex]\stackrel{50}{\sqrt{5300}}[/itex]) = [itex]\Phi[/itex](0.69) = 0.7549

Thanks again :D
 

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