7 Random Variables - M1L3HW1 – Bayesian Statistics

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--- title: "Random Variables - M1L3HW1" subtitle: "Bayesian Statistics: From Concept to Data Analysis" categories: - Bayesian Statistics keywords: - Random Variables - Homework --- ::::: {.content-visible unless-profile="HC"} ::: {.callout-caution} Section omitted to comply with the Honor Code ::: ::::: ::::: {.content-hidden unless-profile="HC"} ::: {#exr-rv-1} [**random variables**]{.column-margin} When using random variable notation, $X$ denotes what? ::: {.solution .callout-tip collapse=true} #### Solution: `random variable`. ::: :::: ::: {#exr-rv-2} [**random variables**]{.column-margin} When using random variable notation, little $x$ or $X=x$ denotes what? ::: {.solution .callout-tip collapse="true"} #### Solution: a **realization of a random variable**. It is a possible value the random variable can take. ::: ::: ::: {#exr-rv-3} When using random variable notation, $X \sim$ denotes what? ::: ::: {.solution .callout-tip collapse="true"} #### Solution: **distributed as**. ::: ::: {#exr-rv-4} What is the value of $f(x)=−5\mathbb{I}_{x>2}(x)+x\mathbb{I}_{x<−1}(x)$ when $x=3$? ::: ::: {.solution .callout-tip collapse="true"} #### Solution: 5 Only the first term is evaluated as non-zero. ::: ::: {#exr-rv-5} What is the value of $f(x)=−5\mathbb{I}_{x>2}(x)+x \mathbb{I}_{x<−1}(x)$ when $x=0$? ::: ::: {.solution .callout-tip collapse="true"} #### Solution: 0 as all indicator functions evaluate to zero. ::: ::: {#exr-rv-6} Which of the following scenarios could should one model with a Bernoulli RV? ::: ::: {.solution .callout-tip collapse="true"} #### Solution: Predicting whether your hockey team wins its next game (tie counts as a loss) ::: ::: {#exr-rv-7} Calculate the expected value of the following random variable: $X$ takes on values $\{0, 1, 2, 3\}$ with corresponding probabilities $\{0.5, 0.2, 0.2, 0.1\}$. ::: ::: {.solution .callout-tip collapse="true"} #### Solution: $0(.5)+1(.2)+2(.2)+3(.1) =0.9$ ::: ::: {#exr-rv-8} Which of the following scenarios could we appropriately model using a binomial RV (with n \> 1)? ::: ::: {.solution .callout-tip collapse="true"} #### Solution: Predicting the number of wins in a series of three games against a single opponent (ties count as losses) ::: ::: {#exr-rv-9} If $X\sim Binomial(3,0.2)$. Calculate $\mathbb{P}r(X=0)$. ::: ::: {.solution .callout-tip collapse="true"} #### Solution: ```{python} import math def binom(n,p,x): return round(math.comb(n,x) * p**x * (1-p)**(n-x),2) print(f'{binom(n=3,p=0.2,x=0)=}') ``` ::: ::: {#exr-rv-10} If $X\sim Binomial(3,0.2)$. Calculate $\mathbb{P}r(X \le 2)$. ::: ::: {.solution .callout-tip collapse="true"} #### Solution: ```{python} print(f'{binom(n=3,p=0.2,x=0)+binom(n=3,p=0.2,x=1)+binom(n=3,p=0.2,x=2)}') ``` ::: :::::