Normal Distribution

Determine the probability $P\left(T<14\right)$ without technology, for $T~N\left(17.1, 3.1^2\right)$.

Estimate the probability that for a randomly selected commuter train it will take at least $214$ seconds for all passengers to board if the data is collected from a large number of rush hour commuter trains. The time $T$ for all the passengers to board a train is distributed normally with mean $186$ seconds and standard deviation $14$ seconds.

A national park houses a large number of tall trees. The heights of the trees are normally distributed with mean of $45$ metres and a standard deviation of $9$ metres. One tree is selected at random from the park. Given that this tree is at least $55$ metres, find the probability that it is taller than $65$ metres.

Use the standard values:

$$\begin{gathered} P\left(0<Z\le 2.22\right)=0.4868 \\ P\left(0<Z\le 1.11\right)=0.3665 \end{gathered}$$

Let $X~N\left(35, 4^2\right).$ Find: $P(X\ge 46.5)$

Let $X~N\left(35, 4^2\right).$ Find: $P(40.5⩽X<46.5)$

Let $X~N\left(35, 4^2\right).$ Find: $P(30.5⩽X<40.5)$

Let $X~N\left(35, 4^2\right).$ Find: $P\left(X<30.5\right)$

Suppose$r.v. \mathrm{X} =$ Waiting time in minutes for a bus by a passenger and its p.d.f. is given by

$f\left(x\right)=\left\{\begin{array}{ll}\frac{1}{5},& 0\le x\le 5\\ 0& \text{ otherwise }\end{array}\right.$, Find the probability that waiting time is more than$4 \text{minutes}.$

Suppose random variable$\left\{X\right\}=$ Waiting time in minutes for a bus by a passenger and its probability density function is given by $f\left(x\right)=\left\{\begin{array}{ll}\frac{1}{5},& 0\le x\le 5\\ 0& \text{ otherwise }\end{array}\right.$. Find probability that waiting time is between$1$and $3$minutes, 

Suppose the error involved in making a certain measurement is a continuous $r. v. \mathrm{X}$ with p.d.f.

$f\left(x\right)=\left\{\begin{array}{ll}k\left(4-x^2\right),& -2\le x\le 2\\ 0,& \mathrm{otherwise}\end{array}\right.$

Compute $P(X<-0.5\text{ or }X>0.5)$.

Suppose the error involved in making a certain measurement is a continuous $r. v. \mathrm{X}$ with p.d.f.

$f\left(x\right)=\left\{\begin{array}{ll}k\left(4-x^2\right),& -2\le x\le 2\\ 0,& \mathrm{otherwise}\end{array}\right.$

Compute $P(-1<X<1)$   [Write your answer as lowest fraction form]

Suppose the error involved in making a certain measurement is a continuous $r. v. \mathrm{X}$ with p.d.f.

$f\left(x\right)=\left\{\begin{array}{ccc}k\left(4-x^2\right)& ,& -2\le x\le 2\\ 0& ,& \text{ otherwise. }\end{array}\right.$

$P(X>0)=\frac{a}{b}$. Find $a+b$.

Let $X=$ time (in minutes) that elapses between the bell and the end of the lecture in case of a college professor. Suppose $X$ has p.d.f.

$f\left(x\right)=\left\{\begin{array}{lll}k{x}^2& ,& 0\le x\le 2\\ 0& ,& \mathrm{otherwise}\end{array}\right.$. If $k=\frac{a}{b}$, then find the value of $a+b$.

Let $f\left(x\right)=\left\{\begin{array}{ll}\frac{{\displaystyle 1}}{{\displaystyle x^2}}& 1<x<\mathrm{\infty };\\ 0,& \mathrm{otherwise}\end{array}\right.$  be the probability density function of a random variable $X$.  If $C_1=\{x:1<x<2\}$ and $C_2=\{x:4<x<5\}$, find $P\left(C_1\cup C_2\right)$. (Write the answer in decimal only)

For the following p.d.f.s of $X$, find
$P\left(X<1\right)$ and $P\left(\left|X\right|<1\right)$

$f\left(x\right)=\left\{\begin{array}{ccc}\frac{x+2}{18}& ,& -2<x<4;\\ 0& ,& \mathrm{otherwise}\end{array}\right.$

For the following p.d.f.s of $X$, find
$P\left(X<1\right)$ and $P\left(\left|X\right|<1\right)$

$\begin{array}{r}f\left(x\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\frac{x^2}{18},\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-3<x<3\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=0,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{otherwise}\end{array}$