Random Variables

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  • Q1

    Define mean of a random variable.

    Marks:1
    Answer:

    Suppose a random variable X assumes the values x1, x2, ..., xn with probabilities of occurrence p1, p2, ..., pn respectively. Then the mean of the random variable X is defined as

    E(x) = x1p1 + x2p2+... xnpn = ∑xp.

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  • Q2

    Two bad eggs are mixed accidentally with 10 good one. Find the probability distribution of one bad egg in three, drawn at random, from this lot.

    Marks:1
    Answer:

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  • Q3

    Check whether the following can be probability distribution of a random variable X:

    X: 0 1 2
    P(X): 0.3 0.5 0.1

    Marks:1
    Answer:

    P(X = 0) + P(X = 1) + P(X = 2) = 0.3 + 0.5 + 0.1
    = 0.9 ≠ 1

    Hence, the given distribution of probabilities is not a probability distribution.

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  • Q4

    Define random variable.

    Marks:1
    Answer:

    A random variable is a real valued function having domain as the sample space associated with a random experiment.

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  • Q5

    If X is a random variable and k is a constant then prove that

    Var(kX) = k2VarX.

    Marks:1
    Answer:

    Var(X) = E(X2) – [E(X)]2

    Var(kX) = E(k2X2) – [E(kX)]2

    = k2E(X2) – [kE(X)]2

    = k2E(X2) – k2[E(X)]2

    = k2[E(X2) – {E(X)2}]

    = k2Var(X)

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