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Math Help - Proof by Induction (Urgent hint required)

  1. #1
    Senior Member slevvio's Avatar
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    Proof by Induction (Urgent hint required)

    Let X be a discrete random variable and  g: R_X \rightarrow \mathbb{R} a continuous function, which is convex, i.e. for all  x_1,x_2 \in R_X and  \lambda \in (0,1)

     g(\lambda x_1 + (1-\lambda)x_2 \leq \lambda g(x_1) + (1-\lambda) g(x_2).

    (a) For  \lambda_i \geq 0 with  \sum_{i=1}^{n} \lambda_i = 1 show that  g(\sum_{i=1}^{n} \lambda_i x_i ) \leq \sum_{i=1}^{n} \lambda_i g(x_i)

    I have let P(n) denote the above statement and shown that it holds for n = 1, but I am finding it hard to progress much further. I really need help with this and any hints or tips would be appreciated.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by slevvio View Post
    Let X be a discrete random variable and  g: R_X \rightarrow \mathbb{R} a continuous function, which is convex, i.e. for all  x_1,x_2 \in R_X and  \lambda \in (0,1)

     g(\lambda x_1 + (1-\lambda)x_2 \leq \lambda g(x_1) + (1-\lambda) g(x_2).

    (a) For  \lambda_i \geq 0 with  \sum_{i=1}^{n} \lambda_i = 1 show that  g(\sum_{i=1}^{n} \lambda_i x_i ) \leq \sum_{i=1}^{n} \lambda_i g(x_i)

    I have let P(n) denote the above statement and shown that it holds for n = 1, but I am finding it hard to progress much further. I really need help with this and any hints or tips would be appreciated.
    Google for Jensen's inequality.

    CB
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  3. #3
    Senior Member slevvio's Avatar
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    Thank you, managed to solve it
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