Probability Distribution of Continuous Random Variables

The continuous random variable $X$ represents the amount of sunshine in hours between noon and $4$ pm at a skiing resort in the high season. The probability density function, $\mathrm{f}\left(x\right)$, of $X$ is modelled by $f\left(x\right)=\left\{\begin{array}{l}\frac{3x^2}{64}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ for }0⩽x⩽4\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ otherwise }\end{array}\right.$. Find $P(2<x<4)$.

Let$f\left(x\right)$  be a continuous random variable with the following probability density function: $f\left(x\right)=\frac{1}{2}\left(x+1\right)$ for $-1<x<1$. Then the ${64}^{\mathrm{th}}$ percentile is _____.

Find the quartiles of the following data: $4, 6, 7, 8, 10, 23, 34$.

If probability density function of a continuous random variable $\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{ll}\mathrm{\{}-{\mathrm{x}}^2+2\mathrm{x}-16,& 0<\mathrm{x}<2\\ 0,& \mathrm{otherwise}\end{array}\right.$, then the median of the distribution is 

Let $X$ be a continuous random variable with the PDF given by: $f\left(x\right)=\left\{\begin{array}{r}x; 0<x<1\\ 2-x; 1<x<2\\ 0; x>2\end{array}\right.$. Find $P(0.5<x<1.5)$.

Let $f\left(x\right)$ be a continuous random variable with the following probability density function: $f\left(x\right)=\frac{1}{2}\left(x+1\right)$ for $-1<x<1$. What is the ${64}^{\mathrm{th}}$ percentile of ?

Find the first two quartiles of the following data: $4, 6, 7, 8, 10, 23, 34$.

Given the following probability density function of a continuous random variable, then prove that the median of the distribution is $1$.

$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{ll}\mathrm{\{}-{\mathrm{x}}^2+2\mathrm{x}-16,& 0<\mathrm{x}<2\\ 0,& \mathrm{otherwise}\end{array}\right.$

Samuel buys a bag of kindling. What is the expected mass of kindling Samuel will get in the bag?($K=\frac{1}{70}$) if the mass of kindling in a bag, $\mathrm{X} \mathrm{kg}$, is a continuous random variable with probability density function given by:

$\mathrm{f}\left(\mathrm{x}\right) = \left\{\begin{array}{ll}\mathrm{K}(3{\mathrm{x}}^2-4\mathrm{x}+2)& 3\le \mathrm{x}\le 5\\ 0& \mathrm{otherwise}\end{array}\right.$

Find the mean and variance of $\mathrm{X}$ which has a continuous random variable,$\mathrm{X}$ , with probability density function as follows:

$f\left(x\right)=\left\{\begin{array}{ll}\frac{6}{125} \left(5\mathrm{x}-{\mathrm{x}}^2\right)& 0\le \mathrm{x}\le 5\\ 0& \mathrm{otherwise}\end{array}\right.$

A continuous random variable,$\mathrm{X}$ , has probability density function as follows:

$f\left(x\right)=\left\{\begin{array}{ll}\frac{6}{125} \left(5\mathrm{x}-{\mathrm{x}}^2\right)& 0\le \mathrm{x}\le 5\\ 0& \mathrm{otherwise}\end{array}\right.$

Find the mean and variance of the continuous random variable $\mathrm{X}$.

A continuous random variable, $X$, has the following probability function:

$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}k\left(x-2\right) 2⩽x⩽5\\ k\left(7-x\right) 5⩽x⩽7\\ 0 \mathrm{otherwise}\end{array}\right.$

Here, $k=\frac{2}{13}$.

Find the value of $x$ in surd, such that $P\left(X<x\right)=20\%$.

A continuous random variable, $X$, has the following probability function:

$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}\frac{2}{25}\left(5-x\right) 0⩽t⩽5\\ 0 \mathrm{otherwise}\end{array}\right.$

The lifetime, in years, of an electrical component is the random variable $T,$ with probability density function $f$ given by

$f\left(t\right)=\left\{\begin{array}{l}A{e}^{-\lambda t} t\ge 0,\\ 0 \mathrm{otherwise},\end{array}\right.$

where $A$ and $\lambda$ are positive constants. It is known that out of $100$ randomly chosen components, $16$ failed within the first year.

Find an estimate for the value of $\lambda ,$ and hence find an estimate for the median value of $T.$

The lifetime, in years, of an electrical component is the random variable $T,$ with probability density function $f$ given by

$f\left(t\right)=\left\{\begin{array}{l}A{e}^{-\lambda t} t\ge 0,\\ 0 \mathrm{otherwise},\end{array}\right.$

where $A$ and $\lambda$ are positive constants. Show that $A=\lambda .$

The cumulative distribution function of a continuous random variable $Y$ is $g\left(y\right)=\left\{\begin{array}{ll}\frac{{\displaystyle 1}}{{\displaystyle 16}}& 0⩽y⩽4\\ \frac{{\displaystyle 1}}{{\displaystyle 8\sqrt{y}}}& 4⩽y⩽25\\ 0& \text{otherwise}\end{array}\right.$. Show that $E\left(Y\right)=10.25$

The time, $T$ seconds, between successive cars passing a particular checkpoint on a wide road has probability density function $f$ given by $\mathrm{f}\left(\mathrm{t}\right)=\left\{\begin{array}{l}\frac{1}{100}{e}^{-0.01t } t⩾0\\ 0 \mathrm{otherwise}\end{array}\right.$

State the expected value of $T.$

The continuous random variable $X$ has the following probability density function:

$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}k{x}^4+\frac{1}{5} 0<x<2\\ 0 \mathrm{otherwise}\end{array}\right.$

Sketch the graph of $\mathrm{f}\left(\mathrm{x}\right)$

The continuous random variable $X$ has the following probability density function:

$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}k{x}^4+\frac{1}{5} 0<x<2\\ 0 \mathrm{otherwise}\end{array}\right.$

  Show that $k=\frac{3}{32}$. 

A random variable, $X$, has probability density function given by:

$f\left(x\right)=\left\{\begin{array}{l}k\sin x\\ 0\end{array} \begin{array}{c}0⩽x⩽\frac{2}{3}\pi \\ \mathrm{otherwise}\end{array}\right.$

where $k$ is a constant, and $k=\frac{2}{3}$ as shown in the diagram.

Find $E\left(X\right)$.