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Thread: independent random variables

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    independent random variables

    Suppose that $\displaystyle A, B, C,$ are independent random variables, each being uniformly distributed over $\displaystyle (0,1).$

    (a) What is the joint cumulative distribution function of $\displaystyle A, B, C?$
    (b) What is the probability that all the roots of the equation $\displaystyle Ax^2 + BX + C = 0$ are real?
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    Hello,

    Let x,y,z in $\displaystyle \mathbb{R}$

    The cdf of (A,B,C) is $\displaystyle P(A\leq x,B\leq y,C\leq z)=P(\{A\leq x\}\cap \{B\leq y\} \cap \{C\leq z\})$

    But since A,B,C are independent, the probability of the intersection is the product of the probabilities

    So the cdf is $\displaystyle P(A\leq x)P(B\leq y)P(C\leq z)$
    And since you certainly know the cdf of a uniform distribution, you'll be able to answer the question...


    You have to find $\displaystyle P(B^2-4AC\geq 0)=\int_0^1\int_0^1 \int_{2\sqrt{ac}}^1 db ~da ~dc$

    Try to understand that...
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