diff --git a/_problems/unit-02/D-cdf-of-a-discrete-RV/1.md b/_problems/unit-02/D-cdf-of-a-discrete-RV/1.md index 08804f6..1c252bd 100644 --- a/_problems/unit-02/D-cdf-of-a-discrete-RV/1.md +++ b/_problems/unit-02/D-cdf-of-a-discrete-RV/1.md @@ -8,7 +8,7 @@ statement: | 3. Provide in bracket notion the cumulative distribution function (cdf) of X 4. What are the probailities P(X <= 2), P(X <= 1.75) and P(X >= 2.3) -Problem modified afterHogg, McKean and Craig - Introduction to Mathematical Statistics + Problem modified afterHogg, McKean and Craig - Introduction to Mathematical Statistics --- Let $Y$ be a bernoulli random variable defined such that each flip @@ -20,8 +20,8 @@ $$ $$ P(Y = y) = \begin{cases} -0.5, &y = 0 \\ -0.5, &y = 1 \\ +0.5, &y = 0 \\\\ +0.5, &y = 1 \\\\ 0, & \text{otherwise} \end{cases} $$ @@ -79,16 +79,15 @@ $$ That is we sum all the mass from $\( -\infty \)$ to desired value of $x$ and recall that this cumulative probability quantity only changes at values of $X$ for which there is positive mass. Since as we recall probability is only in the range of 0 to 1, and recalling that cdf increases at increasing supp values, we have... $$ -\(F_{X}(x) = \left\\{ -\begin{array}{ll} -0 & \text{if } x < 0 \\ -0.0625 & \text{if } 0 \leq x < 1 \\ -0.3125 & \text{if } 1 \leq x < 2 \\ -0.6875 & \text{if } 2 \leq x < 3 \\ -0.9375 & \text{if } 3 \leq x < 4 \\ +F_{X}(x) = +\begin{cases} +0 & \text{if } x < 0 \\\\ +0.0625 & \text{if } 0 \leq x < 1 \\\\ +0.3125 & \text{if } 1 \leq x < 2 \\\\ +0.6875 & \text{if } 2 \leq x < 3 \\\\ +0.9375 & \text{if } 3 \leq x < 4 \\\\ 1 & \text{if } 4 \leq x \\ -\end{array} -\right. \) +\end{cases} $$ For part 4 we can just read off of the cdf in bracket notation