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    4. This is simply an exercise in using the definitions and tables of standard normal, \chi^2
    and t distributions.
    The random variables X_1, X_2, X_3 are independent and identically distributed
    as X_i ~ N(0, 4) for i = 1, 2, 3. Find the following probabilities:

    a. P(X_1+ X_2+ X_3 < 3)
    b. P(X_1^2 < 5)
    c. P(X_1^2+ X_2^2+ X_3^2 < 7)
    d. P(\frac{X_1}{\sqrt(X_2^2+X_3^2)}> 2)

    Part (a) I can do. Part (b) i'm having trouble with as the normal distribution is isn't a standard normal so i'm not sure how to use the \chi^2 distribution.

    Many thanks
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    take square root on both sides and remember that it could be negative
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    (Original post by danny111)
    take square root on both sides and remember that it could be negative
    thanks . any ideas how i would use \chi^2 for part (c) and t distribution for part (d).
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    (Original post by JasdeepKochhar)
    thanks . any ideas how i would use \chi^2 for part (c) and t distribution for part (d).
    c) they are i.i.d. the normals. you know chi squared with n degrees of freedom is the sum of n standard normal squared. how can you rearrange the {X_1}^2 such that they are standard normal squared? {X_1}/2 is standard normal so you want to divide the {X_i}^2 by 4 (so divide by 4 on both sides) then they are square of standard normal. so together with i.i.d. this is \chi^2 with 3 DoF

    d) really the same just use definition of t and not chi
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    (Original post by danny111)
    c) they are i.i.d. the normals. you know chi squared with n degrees of freedom is the sum of n standard normal squared. how can you rearrange the {X_1}^2 such that they are standard normal squared? {X_1}/2 is standard normal so you want to divide the {X_i}^2 by 4 (so divide by 4 on both sides) then they are square of standard normal. so together with i.i.d. this is \chi^2 with 3 DoF

    d) really the same just use definition of t and not chi
    Thanks SO much
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    so together with i.i.d. this is \chi^2 with 3 DoF

    [/QUOTE]

    Can you please explain to me in detail, please ? for part c and d.
    I still don't understand, how you derive chi^2 with 3 DoF. How do you know is 3 DoF ?
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    Can you please explain to me in detail, please ? for part c and d.
    I still don't understand, how you derive chi^2 with 3 DoF. How do you know is 3 DoF ?
    If you watch the first 10 minutes or so of the Review Session on Jan 27th, James goes through some of these basic results
 
 
 
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