Anonymous
Anonymous asked in Science & MathematicsMathematics · 6 years ago

Help with Uniform Probability Function from 0 to 10?

I am having trouble understanding this for my high school stats class. I need to find the probabilities for the following (an explanation on how to get the probability would be appreciated).

P(0.8 < X < 8.0)

P(0.2 < X < 4.0 or 5.6 < X < 7.2)

P(X = 2)

P(-1 < X < 9.0)

Again it just says consider a continuous probability distribution from 0 to 10, what are the following probabilities?

1 Answer

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  • 6 years ago
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    Well,

    just have a look at this notion of "indication of set" notion

    a random variable X that is uniformely distributed

    over an interval J = [a , b] has a pdf

    ----------------> f_x (x) = [1/ (b-a) ] . 1I_J (x)

    where : 1I_J is the indication function of interval J = [a,b]

    defined by :

    1I_J (x) = 1 , if x € J and

    1I_J (x) = 0 else.

    therefore :

    for any measurable subset A of R :

    P( X € A ) = Integral ( x € A ) f_x (x) dx

    and

    here,

    according to the definition of f_x,

    we obviously get (considering the notion of 1I_J )

    P( X € A ) = Integral ( x € A n J) dx

    finally, if you have understood all this (this is what i hope !!)

    we have : a = 0 and b = 10, so 1/(b-a) = 1/10

    we get :

    for each question the associated A subset :

    P(0.8 < X < 8.0) :

    A = (0.8 , 8) so A n J = (0.8 , 8) n [0 , 10] = (0.8 , 8)

    therefore :

    P(0.8 < X < 8.0) = Int (x € (0.8 , 8) ) (1/10) dx = 7.2/10 = 0.72

    now a bit faster A = (0.2 , 4) U (5.6 , 7.2)

    P(0.2 < X < 4.0 or 5.6 < X < 7.2) = (1/10)[ (4- 0.2) + (7.2 - 5.6)]

    now this one is important for your understanding well,

    here, A = {2} is a set of measure (length...) = 0 within R

    therefore, the integral on it is = 0

    and the important notion to understand is :

    in continuous probabilities, the probability that a continuous variable takes a precise value (or a set of measure = 0)

    is = 0,

    so

    P(X = 2) = 0

    here A = (-1 , 9)

    A n J = (0 , 9)

    P(-1 < X < 9.0) = 9/10 = 0.9

    et voilà !

    cheers from southern France !

    hope it' ll help !!

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