Probability related to cumulative distribution function

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The discussion revolves around calculating probabilities related to a cumulative distribution function (CDF) for a discrete random variable. Initial attempts to define P(X ≤ i) and P(X < i) were deemed incorrect due to missing probabilities for integers between 0 and i. A revised approach clarified that P(X < i) can be expressed as 0 for i=0 and as (i-1)^2/N^2 for i>0, while P(X = i) is defined as (2i-1)/N^2 for i>0. The conversation also explored the use of the Heaviside step function and delta functions for a more generalized probability density function. The final consensus emphasizes the importance of ensuring that the sum of probabilities equals 1 and accurately represents the distribution.
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Homework Statement
Please see below
Relevant Equations
F(X) = P(X ≤ x)
1697385605593.png


I have tried to answer all the questions but I am not that sure with my answer.

1697386871082.png

That's the graph of ##F_X (x)## (I think)

(i) P (X ≤ i) = ##\frac{i^2}{N^2}## and P(X < i) = 0
All of these are based on the graph

(ii) P(X = i) = P(X ≤ i) - P(X < i) = ##\frac{i^2}{N^2}##

Are my answers correct? Thanks
 
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songoku said:
Homework Statement: Please see below
Relevant Equations: F(X) = P(X ≤ x)

View attachment 333620

I have tried to answer all the questions but I am not that sure with my answer.

View attachment 333621
That's the graph of ##F_X (x)## (I think)

(i) P (X ≤ i) = ##\frac{i^2}{N^2}## and P(X < i) = 0
All of these are based on the graph
If there are several integers between 0 and i, they have positive probability values. so P(X<i) > 0.
songoku said:
(ii) P(X = i) = P(X ≤ i) - P(X < i) = ##\frac{i^2}{N^2}##

Are my answers correct? Thanks
No. Notice that your diagram only has one i < N, but there might be several others. Also, the sum of the probabilities must equal 1, so your diagram is missing a lot of probability.
 
FactChecker said:
If there are several integers between 0 and i, they have positive probability values. so P(X<i) > 0.

No. Notice that your diagram only has one i < N, but there might be several others. Also, the sum of the probabilities must equal 1, so your diagram is missing a lot of probability.
Ah, I see. Now I understand the question

Revised attempt:
(i)
$$P (X ≤ i) = \frac{i^2}{N^2}$$

$$P(X < i) =
\begin{cases}
0 & \text{if } i= 0 \\
\frac{(i-1)^2}{N^2} & \text{if } i>0
\end{cases}
$$

(ii)
$$P(X = i) =
\begin{cases}
0 & \text{if } i= 0 \\
\frac{2i-1}{N^2} & \text{if } i>0
\end{cases}
$$

For P(X < i) and P(X = i), is there an answer not involving piecewise function? Thanks
 
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just resuming your effort : using Heaviside step function H
F_X(x)=\frac{1}{N^2}\sum_{i=1}^N (i^2-(i-1)^2)H_1(x-i)
Probability density is by differentiation
p(x)=\frac{1}{N^2}\sum_{i=1}^N (i^2-(i-1)^2)\delta(x-i)
Probability for digits are by integrating p(x) around x=i
P(i)=\frac{i^2-(i-1)^2}{N^2}=\frac{2i-1}{N^2}
for ##1 \leq i \leq N##. Otherwise p(x)=0, P(i)=0.
 
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Thank you very much for the help and explanation FactChecker and anuttarasammyak
 
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