Question

Let$c>0\mathrm{and}0\le \alpha \le 1.$Also let X, Y, and Tbe random variables.

(a) If $P(X>c)=\alpha$, determine $P(X\le c)$ in terms of α.

(b) Suppose that $P(Y>c)=\frac{\alpha }{2}\mathrm{and}P(Y<-c)=P(Y>-c)$. Determine $P(-c\le Y\le c)$ in terms of α.

(c) Suppose that $P(-c\le T\le c)=1-\alpha$and also suppose that $P(T<-c)=P\left(T>c\right)$. Find$P\left(T>c\right)$ in terms of α.

Short answer

Part (a)$P(X\le c)=1-\alpha$.

Part (b) $P(-c\le Y\le c)=1-\alpha$.

Part (c) localid="1651633688867" $P(T>c)=\frac{\alpha }{2}$.

Step-by-step solution

Part (a) Step 1. Given information

$P(X>c)=\alpha$

Part (a) Step 2. Solution

$$\begin{gathered} P(X\le c)=1-P(X>c) \\ =1-\alpha \end{gathered}$$

Part (b) Step 1. Given information 

$P(Y>c)=\frac{\alpha }{2}\mathrm{and}P(Y<-c)=P(Y>-c)$

Part (b) Step 2. Solution

$$\begin{gathered} P(-c\le Y\le c)=1-\left[P\right(Y<-c)+P(Y>-c\left)\right] \\ =1-\left[2P\right(Y<-c\left)\right] \\ =1-\left[\frac{2\alpha }{2}\right] \\ =1-\alpha \end{gathered}$$

Part (c) Step 1. Given information

$P(-c\le Y\le c)=1-\alpha andP(T>c)=P(T<-c)$

Part (c) Step 2. Solution

$$\begin{gathered} P(T>c)=1-[P(-c\le Y\le c)+P(T<-c\left)\right] \\ P(T>c)=1-[1-\alpha +P(T>c\left)\right] \\ 2P(T>c)=\alpha \\ P(T>c)=\frac{\alpha }{2} \end{gathered}$$