High school probability Uncovered: Discover the Secrets with Plainmath's Expert Help

Recent questions in High school probability
High school probabilityAnswered question
Nigro6f Nigro6f 2022-10-18

Is responses in statistics the equivalent to random variables in probability?
The focus of this class is multivariate analysis of discrete data. The modern statistical inference has many approaches/models for discrete data. We will learn the basic principles of statistical methods and discuss issues relevant for the analysis of Poisson counts of some discrete distribution, cross-classified table of counts, (i.e., contingency tables), binary responses such as success/failure records, questionnaire items, judge's ratings, etc. Our goal is to build a sound foundation that will then allow you to more easily explore and learn many other relevant methods that are being used to analyze real life data. This will be done roughly at the introductory level of the first part of the required textbook by A. Agresti (2013), which covers a superset of A. Agresti (2007)
in which, is responses here (statistics) the equivalent to random variables in probability
another page in that site says
Discretely measured responses can be:
Nominal (unordered) variables, e.g., gender, ethnic background, religious or political affiliation
Ordinal (ordered) variables, e.g., grade levels, income levels, school grades
Discrete interval variables with only a few values, e.g., number of times married
Continuous variables grouped into small number of categories, e.g., income grouped into subsets, blood pressure levels (normal, high-normal etc)
We we learn and evaluate mostly parametric models for these responses.
are variables and responses interchangeable here?

High school probabilityAnswered question
Lara Cortez Lara Cortez 2022-10-17

Confusion about sum of random variables conditional probabilities
Let X,Y be independent random variables, and Z = X + Y .. Then I want to calculate P r [ X = x Z = z ] .. My confusion is on evaluating this expression. On the one hand, I have
P r [ X = x Z = z ] = P r [ Z Y = x Z = z ] = P r [ z Y = x ] = P r [ Y = z x ] .
But also, P r [ Z = z X = x ] = P r [ X + Y = z X = x ] = P r [ Y = z x ] . So these two probabilities are equal? But P r [ Z = z X = x ] P r [ X = x ] = P r [ X = x Z = z ] P r [ Z = z ] and in general P r [ X = x ] and P r [ Z = z ] are not equal. I believe it should be P r [ X = x Z = z ] = P r [ Y = z x Z = z ] but I don't think the conditional Z = z can be removed since Y and Z are not independent?
I'm not sure whether the first equation holds either. For example, if I roll a fair six sided die X (numbered 1 to 6) and roll a fair ten sided die Y and take the sum, then P r [ X = 1 Z = 2 ] = 1 since the only possible outcome is ( x , y ) = ( 1 , 1 ) ,, and this is not equal to P r [ Y = ( 2 1 ) ] = 1 / 10.. On the other hand it is equal to P r [ Y = ( 2 1 ) Z = 2 ] = 1.. I think I'm making a mistake in one of these but it's not clear to me in which step.
(The context of this was that X is a random variable with given distribution representing some unknown parameter and Y is a standard normal error. Then you observe z = x + y and want to estimate the X.)

High school probability is one of those interesting tasks that young students receive as part of their statistics and probability assignments. You should also find the answers to probability exercises for high school along with the high school probability questions that will help you come up with the most efficient solutions. If you would like to focus on statistical challenges, you should also use high school probability problems as well as it is based on our examples. Be it related to equations or graphs that will talk about probability, follow our examples and things will become clearer!