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Link between a pdf and a pmf

1

$begingroup$

Watching MIT's poisson process lecture. And the lecturer writes a formula:

$$f_{Y_k}(t) delta = P(t leq Y_k leq t+delta)$$

This is awesome. What is a rigorous, general way of writing this? Is it true that for any random variable $X$, its pmf is linked to its pdf by the following?

$$f_X(x) dx = P(x <= X <= x + dx) + o(dx^2)$$

? I just made that up that small o, because I read somewhere that $f_X(x) dx$ is only an approximation of the RHS.

probability

share|cite|improve this question

edited Mar 24 at 13:42

i squared - Keep it Real

asked Mar 24 at 12:48

i squared - Keep it Reali squared - Keep it Real

1,63211128

$endgroup$

add a comment | 

1

$begingroup$

Watching MIT's poisson process lecture. And the lecturer writes a formula:

$$f_{Y_k}(t) delta = P(t leq Y_k leq t+delta)$$

This is awesome. What is a rigorous, general way of writing this? Is it true that for any random variable $X$, its pmf is linked to its pdf by the following?

$$f_X(x) dx = P(x <= X <= x + dx) + o(dx^2)$$

? I just made that up that small o, because I read somewhere that $f_X(x) dx$ is only an approximation of the RHS.

probability

share|cite|improve this question

edited Mar 24 at 13:42

i squared - Keep it Real

asked Mar 24 at 12:48

i squared - Keep it Reali squared - Keep it Real

1,63211128

$endgroup$

add a comment | 

$begingroup$

Watching MIT's poisson process lecture. And the lecturer writes a formula:

$$f_{Y_k}(t) delta = P(t leq Y_k leq t+delta)$$

This is awesome. What is a rigorous, general way of writing this? Is it true that for any random variable $X$, its pmf is linked to its pdf by the following?

$$f_X(x) dx = P(x <= X <= x + dx) + o(dx^2)$$

? I just made that up that small o, because I read somewhere that $f_X(x) dx$ is only an approximation of the RHS.

probability

share|cite|improve this question

edited Mar 24 at 13:42

i squared - Keep it Real

asked Mar 24 at 12:48

i squared - Keep it Reali squared - Keep it Real

1,63211128

$endgroup$

share|cite|improve this question

edited Mar 24 at 13:42

i squared - Keep it Real

asked Mar 24 at 12:48

i squared - Keep it Reali squared - Keep it Real

1,63211128

share|cite|improve this question

edited Mar 24 at 13:42

i squared - Keep it Real

asked Mar 24 at 12:48

i squared - Keep it Reali squared - Keep it Real

1,63211128

asked Mar 24 at 12:48

i squared - Keep it Reali squared - Keep it Real

1,63211128

asked Mar 24 at 12:48

i squared - Keep it Reali squared - Keep it Real

1,63211128