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    The random variable, X, has a probability density function
    f(x) = kx3 between 0 ≤ x ≤ 2 and 0 outside this interval


    a) Find the value of k: 1/4
    b) Find E(X): 1.6
    c) Find Var(X): 0.107
    d) Find the Median of the Distribution: 1.68
    e) Find the probability than an observation lies within one standard deviation of the mean
    f) Find the probability that out of four observations, two are greater than the mean and two are less than the mean

    Need some help with e and f, not too sure how to do it. Any help is appreciated
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    (Original post by Super199)
    The random variable, X, has a probability density function
    f(x) = kx3 between 0 ≤ x ≤ 2 and 0 outside this interval


    a) Find the value of k: 1/4
    b) Find E(X): 1.6
    c) Find Var(X): 0.107
    d) Find the Median of the Distribution: 1.68
    e) Find the probability than an observation lies within one standard deviation of the mean
    f) Find the probability that out of four observations, two are greater than the mean and two are less than the mean

    Need some help with e and f, not too sure how to do it. Any help is appreciated
    e) Within 1 s.d. of the mean means it must lie in the interval [mean - 1 s.d, mean + 1 s.d.]. You just need to integrate the pdf between those two limits, to find the probability it lies in that interval.

    f)
    Work out the probablity that one observation is less than the mean.

    You then have to consider 4 observations.
    The probability that any given observation is less than the mean is a constant.
    I.e. 4 idential trials.
    You can think of less than the mean as a success, and greater than the mean as a failure.
    So, you're looking for 2 successes out of 4 trials, where the probability of a success is a constant.

    What distribution does this suggest to you?
 
 
 
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