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Equally likely values 1 to n: mean and variance

This type of question, asked 3 times: 10 May 2023 S2, 9 May 2024 S2, 12 May 2024 S2. All 3 questions below, oldest first.

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  1. Question 1

    10 May 2023 S2

    123
    A random variable X assumes values 1,2,3,…,n1, 2, 3, \ldots, n with equal probabilities. If Var(X):E(X)=4:1\text{Var}(X) : E(X) = 4 : 1, then n is equal to

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  2. Question 2

    9 May 2024 S2

    148
    A random variable X assumes values 1,2,3,…,n1,2,3,\ldots,n with equal probabilities, if Var(X)=E(X)\text{Var}(X) = E(X), then nn is

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  3. Question 3

    12 May 2024 S2

    122
    A discrete random variable XX can take all possible integer values from 1 to kk, each with a probability 1k\frac{1}{k}, then its variance is

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