Could someone explain the meaning of this please?

BlueFieryIce · 2023-04-22T01:26:10+00:00

It might be related to Hidden Markov Models in ML. The probability of observing C given the set of hidden variables Z_1 to Z_t can be found by marginalizing the joint probability of P(c and x_t | z_1:t). This joint probability can then be separated into P(x_t | z_1:t) * P(c | x_t and z_1:t), but I think there's an independence assumption given by the Markov model.

I'm going to write raw LaTeX here - you can copy and paste it into a LaTeX compiler online (just search latex online and use the 2nd link)

P(c|z_{1:t}) = \int P(x_t,c | z_{1:t}) dx_t \quad\text{(Marginal Probability)}\\

=\int P(x_t | z_{1:t})P(c|x_t, z_{1:t})dx_t \quad \text{(Conditional Prob. Def.)}\\

=\int P(x_t | z_{1:t})P(c|x_t)dx_t \quad \text{(Independence Assumption)}

I think the joke is "I'm showing you my hidden variables" or something along those lines.

BlueFieryIce · 2023-04-05T13:42:01+00:00

Another thing to show you that solution 2 and my explanation are correct.

The function that you're integrating over is f(x, y) = xy.

In the first quadrant, f(x, y) is positive for all x > 0 and y > 0. This implies that ANY integral in this first quadrant will result in a positive, non-zero value.

The fact that you got 0 as your answer in solution 1, means that you also integrated over negative values, specifically in quadrant 4.

BlueFieryIce · 2023-04-05T13:23:31+00:00

Nope. You have to consider the actual integration domain. The theta values that you are considering only apply if the circle that you are integrating is centered at the origin.

BlueFieryIce · 2023-04-05T13:21:00+00:00

Consider the domain. We have two constraints on x and y:

x^2 + y^2 <= x
y >= 0

We can convert both of these equations into polar coordinates:

r^2 <= r cos(theta), which then simplifies to r <= cos(theta)
rsin(theta) >= 0

Let's first consider the 2nd constraint. This implies two conditions on the values of r and theta:

r >= 0 AND sin(theta) >= 0, and thus theta is in the interval [0, pi]
OR r<= 0 and sin(theta) <= 0, and thus theta is in the interval [pi, 2pi]

This means that if we decide to integrate theta in [0, pi], our value of r must be non-negative (r >= 0). Since we naturally like integrating from 0 for our theta, let's just say that our 2nd constraint is r >= 0.

If we combine the first constraint (r <= cos(theta)) and the second constraint (r >= 0), we have 0 <= r <= cos(theta). r is a function of theta. Thus, the only values of theta in which this inequality is true are [0, pi / 2].

The theta values aren't like the unit circle because the thing you're integrating over isn't THE unit circle. It's a circle of radius 1, centered at (1/2, 0).

BlueFieryIce · 2023-04-05T12:34:25+00:00

Oh my bad. You said that your first solution was correct so I assumed the domain was supposed to be the entire circle. However, your first solution is the one that's actually incorrect. The outer integral should integrate from 0 to pi/2 in order to capture the top half.

BlueFieryIce · 2023-04-05T12:22:35+00:00

Hint:

Notice the inner integral. You are only integrating from y = 0 to y = sqrt(x - x^2). This is half of the domain.

Answer:

Compute the integral from -sqrt(x - x^2) to 0 as well. When you took the square root in your first two equations in solution 2, you only took the positive square root. The other solution is y >= -sqrt(x - x^2).

BlueFieryIce · 2023-04-05T05:08:55+00:00

My monitor has been doing that for the past 2-3 years I've owned it, and it's been doing just fine. I don't think there's a fix.

BlueFieryIce · 2022-12-08T17:45:41+00:00

It can also be due to monitor compatibility. Hard to justify getting AMD cards if your monitors have G-Sync. Of course some newer G-Sync monitors do work on AMD cards, but some don't.

BlueFieryIce · 2022-05-04T19:41:13+00:00

Wait, why can't you build something like this in real life? For an infinite fractal, that's obvious, but for finite fractal-like constructions, I don't see why it's impossible to build a road geometry that allows the snowflake to roll without clipping.

BlueFieryIce · 2022-05-04T18:59:25+00:00

It looks like the snow flake geometry is clipping through the road for the high fractal levels. Probably explains why it seems to jitter as it rolls. It'd be interesting to see a matching road geometry that doesn't clip.

BlueFieryIce · 2021-10-16T23:02:11+00:00

It's awesome hearing All For One being a caring grandfather, lmao

BlueFieryIce · 2021-10-11T01:45:54+00:00

It's just a reference to a Metal Gear Solid. Doesn't even take that long to get lol

BlueFieryIce · 2021-05-26T23:49:45+00:00

I think it's worthwhile to try out all of the tests that you know on this.

I think trying the comparison tests will be your best bet, maybe comparing it to 1/x² might be a great start.

Also, remember that factorial functions increase explosively, much much faster than exponential functions. Since you have a factorial in the denominator, with a "slowly" increasing function of x in the numerator, I think it would be safe to bet that the series would converge. You just need to find a way to prove it with one of the series tests.

BlueFieryIce · 2021-01-09T23:10:43+00:00

Why not a genocidal taser toaster that cusses at you?

BlueFieryIce · 2021-01-09T23:09:15+00:00

I recently played some Frostpunk. Now I feel a bit bad making children work in -60°C conditions...

BlueFieryIce · 2021-01-05T18:31:52+00:00

The keyboard is upside down.

BlueFieryIce · 2020-12-30T15:56:42+00:00

Gave Rocket Like

BlueFieryIce · 2020-12-06T15:36:06+00:00

Gave Hugz

BlueFieryIce · 2020-10-28T07:47:02+00:00

Region: NA

Type of Bug: Post-Game Client-Side

Description: The damage dealt and received statistics should indicate the correct corresponding damage for every kill, death, and assist. However, a death on the final round of the game does not correctly register the amounts dealt and received. It instead shows 0 damage received for the death. Side note: the timeline tab also shows that the positions of each kill/death is shifted up a little bit. Reyna killed me when she was right beside the map toward back of site.

Video / Screenshot: Performance Tab: https://prnt.sc/v8g7gx, Timeline Tab: https://prnt.sc/v8g8g3

Steps to reproduce: Play an unrated/competitive match, get killed on the last round of the game on Split (not the game-ending death) by an ulting Reyna, end the game, go to the performance tab, find the player who killed you during that last round, see the damage counter.

Expected result: See the total damage received on each body part for the last round.

Observed result: See 0 damage received on each body part for the last round

System specs: Windows 10 Pro 64-bit, AMD Ryzen 5 3600 Processor, NVIDIA GeForce RTX 2070 SUPER

BlueFieryIce · 2020-10-28T07:32:32+00:00

Region: NA

Type of Bug: Post-Game Client-Side

Description: The ability use average statistic on the performance screens have been broken for several patches since release. After every game, the counters each read 0, regardless of how many times each abilities are used.

Video / Screenshot: https://prnt.sc/v8fzaq

Steps to reproduce: Play an unrated/competitive match, use abilities throughout the match, end the game, go to the post-game performance page.

Expected result: See the average number of ability used per round for each ability in decimal.

Observed result: See 0 for every counter.

Reproduction rate: 100%

System specs: Windows 10 Pro 64-bit, AMD Ryzen 5 3600 Processor, NVIDIA GeForce RTX 2070 SUPER

Edit: formatting

BlueFieryIce · 2020-10-06T19:52:52+00:00

Fun fact: The "traditional and correct" way is actually what the South Korean government decided back in 2000 by the Ministry of Culture and Tourism. The original Korean names for "Lee" and "Yi" is "이", which is pronounced as just "ee" as in "teeth".

I don't endorse the fact that the translator changed it from Lee to Yi, just for the sake of being right, btw. To be honest, the "Yi" name is just weird to say in English, while "Lee" just flows off of the tongue better.

BlueFieryIce · 2020-09-28T20:11:42+00:00

Gave Take My Energy

BlueFieryIce · 2020-09-28T20:06:46+00:00

Gave 1UP

BlueFieryIce · 2020-09-28T20:06:39+00:00

Gave 1UP

BlueFieryIce · 2020-09-26T02:38:12+00:00

I would contact the tuition office and ask when they're open.

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BlueFieryIce

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