Question

Assume that xhas an exponential distribution with$\mathit{\theta }\mathbf{=}\mathbf{3}$.

Find

a.$\mathit{P}\left(x\le 1\right)$

b.$\mathit{P}\mathbf{(}\mathbf{x}\mathbf{>}\mathbf{1}\mathbf{)}$

c.$\mathit{P}\mathbf{(}\mathbf{x}\mathbf{=}\mathbf{1}\mathbf{)}$

d.$\mathit{P}\mathbf{(}\mathbf{x}\mathbf{\le }\mathbf{6}\mathbf{)}$

e.$\mathit{P}\mathbf{(}\mathbf{2}\mathbf{\le }\mathbf{x}\mathbf{\le }\mathbf{10}\mathbf{)}$

Short answer

1. $P\left(x\le 1\right)=0.9503$
   
2. role="math" localid="1660277023394" $P\left(x>1\right)=0.0497$
   
3. role="math" localid="1660277013921" $P\left(x=1\right)=0.1491$
   
4. $P\left(x\le 6\right)=0.999$
   
5. $P\left(2\le x\le 10\right)=0.0024$

Step-by-step solution

Given information

X is an exponential random variable and$\theta =3$ .

Define the probability distribution function

The p.d.f of X is

$$\begin{gathered} f\left(x\right)=\theta e^{-\theta x};x\ge 0 \\ =3e^{-3x} \end{gathered}$$

Calculate

$$\begin{gathered} P\left(x\le 1\right)={\int }_0^{1}3e^{-3x}dx \\ =3{\int }_0^{1}3e^{-3x}dx \\ =3\left(\frac{1}{3}-\frac{e^{-3}}{3}\right) \\ =1-e^{-3} \\ =1-0.0497 \\ =0.9503 \end{gathered}$$

Hence,$P\left(x\le 1\right)=0.9503$$P\left(x\le 1\right)=0.9503$

Calculate P ( x > 1)

$$\begin{gathered} P\left(x>1\right)=1-P\left(x\ge 1\right) \\ =1-3{\int }_0^1{e}^{-3x}dx \\ =1-0.9503 \\ =0.0497 \end{gathered}$$

Hence,$P\left(x>1\right)=0.0497$

Calculate P ( x = 1 ) 

$$\begin{gathered} P\left(x=1\right)=3e^{-3x} \\ =3e^{-3} \\ =3\times 0.0497 \\ =0.1491 \end{gathered}$$

Hence, $P\left(x=1\right)=0.1491$$P\left(x=1\right)=0.1491$

Calculate P(x≤6)

$$\begin{gathered} P\left(x\le 6\right)=3{\int }_0^6{e}^{-3}dx \\ =3\left(\frac{1}{3}-\frac{e^{-18}}{3}\right) \\ =e^{-18}\left(e^{-18}-1\right) \\ =0.999 \end{gathered}$$

Hence,$P\left(x\le 6\right)=0.999$

Calculate

$$\begin{gathered} P\left(2\le x\le 10\right)=3{\int }_2^{10}{e}^{-3x}dx \\ =3\left(\frac{e^{-6}}{3}-\frac{e^{-30}}{3}\right) \\ =e^{-30}\left(e^{-24}-1\right) \\ =0.0024 \end{gathered}$$

Hence,$P\left(2\le x\le 10\right)=0.0024$