$F$ is the cumulative distribution function for a continuous random variable. What is the meaning of $F(b)−F(a)=0.20$?

Question

Does it mean that $[a,b]$ is a length of $0.2$, or that $P(X=b)−P(X=a)=20$% or $P(X∈(a,b])=20$%.

All of these options look plausible to me but I failed to find an explanation to this question online.

It would be interesting to hear the reason(s) for those close-/downvotes. — Peter, Aug 05 '19 at 12:38

Peter Foreman · Accepted Answer · 2019-08-05T12:38:25.757

3

$$F(b)-F(a)=\int_a^bf(x)\,\mathrm{d}x=P(a\lt X\le b)$$ Also we have that $$F(b)-F(a)=P(X\le b)-P(X\le a)=P(a\lt X\le b)$$

edited Aug 05 '19 at 12:38

answered Aug 05 '19 at 12:37

Peter Foreman

Would this mean that you have 20% of getting a value from $a$ to $b$ which includes $b$? – Libor Zachoval Aug 05 '19 at 12:40
Yes exactly that. The probability of a value lying in the interval $(a,b]$ is $20$%. – Peter Foreman Aug 05 '19 at 12:40
and this applies only for continuous distributions correct? – Libor Zachoval Aug 05 '19 at 12:42
1

The second method shows that this also works for discrete distributions. – Peter Foreman Aug 05 '19 at 12:43

drhab · Answer 2 · 2019-08-05T13:11:42.910

2

Let $X$ be a random variable that has $F$ as CDF.

If $a\leq b$ then:$$\{X\leq b\}=\{X\leq a\}\cup\{a<X\leq b\}$$ and both sets are disjoint.

So this leads to:$$F(b)=P(X\leq b)=P(X\leq a)+P(a<X\leq b)=F(a)+P(a<X\leq b)$$or equivalently:$$F(b)-F(a)=P(a<X\leq b)$$

edited Aug 05 '19 at 13:11

answered Aug 05 '19 at 12:43

drhab

2 Answers2