5. Discrete Random Variables I

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Recommends CroCo · 2014-02 · Mathematics Stack Exchange · on “Probability of tossing a coin twice & MATLAB?”

I'm watching this video [this video] at 21:20 min. The guy gave an example for tossing a coin four times. At 23:49 min, he said we can plot the probability of tossing a coin (e.i. $n$) for a large number in MATLAB by using this formula $$ px(k) = \dbinom{n}{k} p^{k}(1-p)^{n-k} \qquad, k=0,1,...,n $$ Ok now I want to plot his example but I'm facing problems. For his example, $n=4 , k = 0,1,2,3,4$ (what does $k=0$ mean here?). This is my code in Matlab clear all clc outcome = 2; % outcome # of a coin (Head & Tail) n = 4; % tossing # k = 0:n; % # of getting head or tail p = 1/n^outcome; % the total probability of all outcomes (1/16) i = 0; % iterator for the loop for i = 0:size(k,2) Px = nchoosek(n,

Recommends Marcus Junius Brutus · 2018-05 · Mathematics Stack Exchange · on “functions of random variables notation”

I understand that a random variable $X$ is a function from the sample space $\Omega$ to $\mathbb{R}$. What I fail to understand is why people when defining a new random variable $Y$, based on some function $g$ that operates on the output of $X$, use the notation $Y=g(X)$. I have seen that both in this site [LINK] and also on the MIT lectures on probability by Prof. Tsitsiklis ( here [this video] on 30:55 and onwards). I think this notation makes no sense at all. The domain of $g$ is $\mathbb{R}$, not some "set of functions from $\Omega$ to $\mathbb{R}$" (which is what the "set of random variables" is). In my view the correct notation ought to be: $$ Y = g \circ X $$ where $\circ$ is function comp

Cites JOHN · 2020-07 · Mathematics Stack Exchange · on “Why random variables is a function? It seems that it violates the definition of function.”

I am watching Lecture 5 of MIT 6.041 Probabilistic Systems Analysis and Applied Probability [this video] . In the lecture, Professor said Random Variable is a function that maps elements from the sample space to a real number. For example, if the sample space is the students in the classroom, a random variable $x$ can be the height of the students. However, I remember that a function is defined as $\forall a \in A \exists! b \in B((a,b) \in F)$ . That means for all inputs, there is one unique output. In the example above, we may have two students with the same height. Will that violate the rule of function? Or Professor just used the word function loosely?