1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Comparing random variables with a normal distribution

  1. Aug 25, 2011 #1
    1. The problem statement, all variables and given/known data

    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. Relevant 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. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 25, 2011 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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]??
     
  4. Sep 1, 2011 #3
    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
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook