I think it's fine to teach continuous probability distributions without explicitly discussing integration. e.g. Using vague statements like "Probabilities are areas under curves, and we can find those areas with mathematical techniques, and software has incorporated those mathematical techniques for us." I think someone can develop a very good understanding of statistics from that perspective. Keep in mind that most of the continuous distributions we use in statistical inference (e.g. normal, t, F, chi-square) do not have closed-form cumulative distribution functions and must be integrated numerically. So whether someone knows integral calculus or not, in the end areas are found using software. For a full and deeper understanding of what's going on? Sure, knowledge of integral calculus is meaningful. At a level to achieve an understanding of applied statistics? I don't think knowledge of integral calculus is necessary.

@@jbstatistics Thanks for the quick reply🙏 I'm actually preparing for an exam, and when I got to the topic of continuous probability distribution, I got a little confused because of the integral. I meant more to solve exam questions related to continuous probability distribution than to have a theoretical understanding. It seems that to reach the final answer, you need to know integral calculus (or have an advanced calculator).

@@Didier-cu6cb For any of those distributions I named, and many others, there is no closed form solution for the integral. It does not matter how great of an integrator one is, the integrals can solved only by numerical techniques. We do that using software. The world's greatest integrator and a person knowing no calculus whatsoever solve the problem in the same way: By asking software for the appropriate area.

@@jbstatistics Right. Thank you for your response & content.

I suggest you limit the "should haves" until you have a deeper understanding of the situation. If your logic here was valid, then there would be no such thing as a two-sided alternative. The value of the statistic will always be on one side or the other, so if we let that determine things we'd always have a one-sided alternative. So your logic can't be valid, or nobody would ever speak of a two-sided alternative. The choice of alternative ******never never never never never ever ever ever ever ever ever**** depends on the value that we see in the sample that we're using to carry out the test. If we do that we're distorting the math and then making false statements in the end. The choice of alternative depends on the nature of the situation at hand. One should be able to write out the appropriate hypotheses without ever looking at the sample data. If the sample data has influenced your choice of hypothesis, then you've violated a fundamental requirement of hypothesis testing.

I think bringing that up in an introductory video on the geometric distribution does more harm than good. It's an extra layer of confusion that people don't need at first. The difference in the random variables, the difference in the means, describing why the variances are the same...it just takes away from the big picture of what the geometric distribution does for us. Sure, I bring it up elsewhere, especially as I use R in my courses and R uses the other definition of the r.v., but I think it would cause more confusion than it's worth in an intro video. Once one is understood, the other comes naturally.

Yes. I'm genuinely telling you they do. Want to compare a two drugs to see which one results in a greater reduction in cholesterol levels? Two surgical methods to see which one results in shorter length of stay? Whether a new diet results in greater weight gain/loss than a standard diet? Investigate whether there is a relationship between happiness levels in society and unemployment rate? Between happiness levels and income? Under certain assumptions and scenarios, these can all be investigated with a t test. The t distribution is fundamental to a t test. Under certain assumptions, often reasonable ones, the t procedures are the best possible procedures to help answer these sorts of questions. I'm a statistician. I'm a statistician because my mind thinks this way, and I want to use information to make the best possible decisions. I have no special love of weird mathematical functions, quirky mathematical exceptions, the vagaries of infinity, or exotic proofs of simple concepts. I'm just a guy trying to get by. This stuff helps our world.

Yes, it is, and I say that and use that there. At that point I had found E(X(X-1)) = lambda^2, and then use that in finding Var(X). Finding E(X(X-1)) is a helpful little trick, as it allows us to cancel the x(x-1) terms from the x! in the denominator of the Poisson pmf. If you try to find E(X^2) directly you will run into problems.

As a general rule I don't use the rejection region approach, even when carrying out a test at a fixed significance level. I think that since the p-value approach leads to the exact same conclusion at a fixed significance level, but gives you the p-value, and the p-value is typically reported by software, that's the better path. Here I was illustrating the difference between t and z tests, in inference for the mean of a normally distributed population. I wasn't getting into the details of hypothesis testing beyond that. It was a fabricated example to show there can be meaningful differences between the t and z, even when the sample size is not all that small. But to address the big picture idea of what you're asking here, in the vast majority of hypothesis testing scenarios no actual decision is being made. Sometimes there is, and if a decision is being made solely on the results of the test (e.g. tossing an explanatory variable out of a multiple regression model in a backwards elimination procedure, say), and so then we need a rule. In the vast majority of cases, no actual decision is being made, and I think it simply doesn't make sense to impose an arbitrary decision rule in spots where no decision is being made. Imposing a binary decision on what is in reality a continuum , where no binary decision is necessary or needed, is not the way to go. Under the assumptions of the test in this video, if H_0 is true then the p-value has a uniform distribution on (0,1). We'd see a p-value of 0.568 or smaller more than half the time if the null is true, so there's no evidence whatsoever against H_0. There doesn't need to be an arbitrary decision rule to come to that conclusion. Much of this is opinion, and we could debate the nuances of hypothesis testing for the next century, as has been done for 1+ centuries. But I'm pretty sure that on my deathbed I'll still think that's a crime against science that everybody got fixated on acting like we're making a reject/don't reject H_0 decision at 0.05 as the default way to go about things.

@jbstatistics I appreciate you taking the time to explain that to me. What a selfless character you have! Respect! I believe that, just like your other videos, this explanation will help many people, not just myself. Thank you again, and keep up the wonderful work! 👏

@@BNT05 Thanks for the kind words! I may not be selfless, but I do truly want to help people learn :)

@@jbstatistics And you manage to do magnificently, imo: 😊

Because as given in the example 20 people are randomly selected.

That would depend on the calculator, and typical calculators do not have the appropriate function. The tables exist because these values are found with software, and people do not always have access to appropriate software. A calculator along the lines of a TI-83 would have a function to do it. A typical $10 calculator will not.