Let $X$: "Launch a die until you get the number 5 for the first time", a discrete random variable. I am asked to calculate $P(10 Is correct to say... If $P(10 ?
How I can calculate $P(10
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probability
probability-theory
probability-distributions
probability-limit-theorems
3 Answers
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$P(10 You need to consider $P(10
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You're correct. Differences are a bit tricky to deal with, but it's clear that
$$P(X \le 10) + P(10 < X \le 20) = P(X \le 20)$$
since the union of the events $\{ X \le 10 \}$ and $\{ 10 < X \le 20 \}$ is just $\{ X \le 20 \}$. Rearranging this gives you
$$P(10 < X \le 20) = P(X \le 20) < P(X \le 10)$$
and if $F_X$ is the cumulative distribution function, with $F_X(x) = P(X \le x)$, then you have
$$P(10 < X \le 20) = F_X(20) - F_X(10)$$
or more generally, when $b \ge a$,
$$P(a < X \le b) = F_X(b) - F_X(a).$$
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Suppose $X$ is always $15$. Then $P(10