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Probabilities in terms of p: range of p and the extreme value of the mean

This type of question, asked 2 times: 9 May 2023 S1, 21 Apr 2025 S1. All 2 questions below, oldest first.

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  1. Question 1

    9 May 2023 S1

    108
    A random variable X takes values −1,0,1,2-1,0,1,2 with probabilities 1+3p4,1−p4,1+2p4,1−4p4\frac{1+3p}{4},\frac{1-p}{4},\frac{1+2p}{4},\frac{1-4p}{4} respectively, where pp varies over R\mathbb{R}. Then the minimum and maximum values of the mean of X are respectively.

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  2. Question 2

    21 Apr 2025 S1

    109
    The following is the probability distribution of X
    XX0123
    P(X=x)P(X = x)1+p5\frac{1 + p}{5}2−2p5\frac{2 - 2p}{5}2−p5\frac{2 - p}{5}2p5\frac{2p}{5}
    For a minimum value of pp, the value of 5E(X)5E(X) is

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