0
$\begingroup$

See this video at 4:30.

enter image description here

According to me, the result should be $1.57$.

$P(X \ge 1.8)$
$ = F(4) - F(1.8) + 1$ (as we have $X>4 = 1$)
$ = 1.57$

Why is the answer of the problem $0.57$, rather than $1.57$?

$\endgroup$
  • 5
    $\begingroup$ You expect the probability to be more than 1? More than certain? $\endgroup$ – badjohn Feb 28 at 13:11
  • $\begingroup$ Why do you add one? I don't understand that. $\endgroup$ – Eff Feb 28 at 13:20
  • $\begingroup$ @Eff, good question. $X \ge 1.8$ means the value of $X$ exceeds $4$. $\endgroup$ – user366312 Feb 28 at 13:22
  • $\begingroup$ @badjohn, $X \ge 1.8$ means the value of $X$ exceeds $4$. So, why not include $1$? $\endgroup$ – user366312 Feb 28 at 13:24
  • $\begingroup$ I considered an answer but others have done so now. $\endgroup$ – badjohn Feb 28 at 13:26
4
$\begingroup$

Yes, the answer should be $0.57$. First and foremost, the answer cannot be more than $1$. Thats a violation of conservation of probability. So the additional $+1$ that you mention in your answer is wrong and not needed, since $F(X>4)=1$ but $P(X>4)=0$. So you need not add the $1$. To prove this statement of mine, note that $$P(X>1.8)=F(X=4)-F(X=1.8)$$ $$=F(X=\infty)-F(X=1.8)$$

It means that the cumulative probability distribution reaches 1, i.e. the sum of all the probabilities reach 1 on crossing X=4. So the cumulative probability function saturates at X=4 with a value of 1. There is no further contribution to F from any point X>4. This implies that P=0 for X>4. And if P=0 in that region, then there is no need for you to add 1 for X>4.

Hope this helps.

$\endgroup$
  • $\begingroup$ "... , since $F(X>4)=1$ but $P(X>4)=0$." --- what do you mean? Kindly rephrase. $\endgroup$ – user366312 Feb 28 at 13:21
  • $\begingroup$ It means that the cumulative probability distribution reaches 1, i.e. the sum of all the probabilities reach 1 on crossing X=4. So the cumulative probability function saturates at X=4 with a value of 1. There is no further contribution to F from any point X>4. This implies that P=0 for X>4. And if P=0 in that region, then there is no need for you to add 1 for X>4. $\endgroup$ – SchrodingersCat Feb 28 at 13:25
  • $\begingroup$ Kindly, add this comment to the main answer so that I can accept this. $\endgroup$ – user366312 Feb 28 at 13:32
  • $\begingroup$ @user366312 Done. $\endgroup$ – SchrodingersCat Feb 28 at 13:41
0
$\begingroup$

The answer is simply $$ \begin{align} P(X\geq 1.8) &= 1-P(X<1.8)\\ &= 1- F(1.8)\\ &= 1-\frac{1}{32}\left(6\cdot1.8^2-1.8^3\right)\\ &\approx 0.57 \end{align} $$

$\endgroup$
  • $\begingroup$ $X \ge 1.8$ means the value of $X$ exceeds $4$. $\endgroup$ – user366312 Feb 28 at 13:24
  • $\begingroup$ @user366312 I must admit that I'm not sure I understand what you're getting at.The value 1.8 does not exceed 4, it's less than 4. But even if it did, I don't understand why it's relevant. $\endgroup$ – Eff Feb 28 at 13:26
  • $\begingroup$ I guess that he is thinking that $X \ge 1.8$ means that $X$ might exceed $4$. Of course, for this particular distribution it won't. $\endgroup$ – badjohn Feb 28 at 13:29
  • $\begingroup$ @badjohn I suppose. But even if could exceed 4, I don't understand how it is relevant. I don't think I understand OP's reasoning. $\endgroup$ – Eff Feb 28 at 13:32
  • $\begingroup$ @Eff I just wondering that it might be the source of his confusion not that I thought that he was right. Clearly he isn't since he calculated a probability of more than $1$. $\endgroup$ – badjohn Feb 28 at 13:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.