For a sample of size $n=3$ from a continuous probability distribution, what is $P(X_{(1)} I'm having trouble trying to solve this question since the median is for the distribution and not the sample. The only explicit formulas for the median I know of are the median $k$ of any random variable $X$ satisfies $P(X≤k)≥1/2$ and $P(X≥k)≥1/2$, but I don't see how to apply that here.
Probability of Median from a continuous distribution
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probability
statistics
2 Answers
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I assume $X_1, X_2, X_3$ are taken to be iid. Here's a hint:
$$P(X_{(1)} < k < X_{(2)}) = 3P(X_1 < k \cap X_2 > k \cap X_3 > k)$$ by a simple combinatoric argument. Do you see why? Since the distributions are continuous, $$P(X_1 > k) = P(X_1 \ge k) = P(X_1 < k) = P(X_1 \le k) = \frac 1 2.$$ The second part of the question is similar.
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1Thanks! That was brilliant, I had so much trouble trying to wrap my head around the idea of comparing the median of a distribution to sample values. – 2011-07-17
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This is also the probability of exactly one success in three trials, with probability $1/2$ of success on each trial. Hence $\binom{3}{1} \left(\frac12\right)^3 = \frac38$.