Discrete and Continuous RV

Q. A continuous random variable X has a probability density function $f\left(x\right)=e^{-x},0<x<\infty .$ Then P{X > 1} is

1. 0.368
2. 0.5
3. 0.632
4. 1.0

Q.

A random variable X has probability density function f(x) as give below:
$f\left(x\right)=(\begin{array}{c}a+\mathrm{bx}\mathrm{for}0<x<1\\ 0\mathrm{otherwise}\end{array}$

If the expected value E[X] = 2/3, then Pr[X < 0.5] is .

1. 0.25

Q. If X is a continuous random variable whose probability density function is given by
$f\left(x\right)=(\begin{array}{cc}k(5x-2x^2),& 0\le x\le 2\\ 0,& \mathrm{otherwise}\end{array}$
Then P(x > 1) is

1. 3/14
2. 4/5
3. 14/17
4. 17/28

Q. If probability density function of a random variable X is $f\left(x\right)=x^2\text{for}-1\le x\le 1\text{and}f\left(x\right)=0$ for any other values of x, then the percentage probability $P(-\frac{1}{3}\le x\le \frac{1}{3})\text{is}$

1. 0.247
2. 2.47
3. 24.7
4. 247

Q. A continuous random variable X has a probability density function

$f\left(x\right)=e^{-2x},0<x<\infty .$

Then P[X > 1] is

1. 0.270
2. 0.067
3. 0.034
4. 0.135

Q.

Let X be a random variable with probability density function
$f\left(x\right)=(\begin{array}{cc}0.2,& \mathrm{for}\left|x\right|\le 1\\ 0.1,& \mathrm{for}1<\left|x\right|\le 4\\ 0,& \mathrm{otherwise}\end{array}$

The probability P(0.5 < X < 5) is .

1. 0.45

Q. A random variable X has the density function $f\left(x\right)=K\frac{1}{1+x^2},\text{where}-\infty <x<\infty .$ Then the value of K is

1. $\pi$
2. $\frac{1}{\pi }$
3. $2\pi$
4. $\frac{1}{2\pi }$

Q. Assume that the duration in minutes of a telephone conversation follows the exponential distribution $f\left(x\right)=\frac{1}{5}{e}^{-x/5},x\ge 0.$ The probability that the conversation will exceed five minutes is

1. $\frac{1}{e}$
2. $1-\frac{1}{e}$
   
3. $\frac{1}{e^2}$
4. $1-\frac{1}{e^2}$

Q.

$\text{Let}{X}_1,X_2,X_3\text{and}{X}_4$ be independent normal random variables with zero mean and unit variance. The probability that $X_4$ is the smallest among the four is .

1. 0.25

Q. The life of a bulb (in hours) is a random variable with an exponential distribution $f\left(t\right)=\alpha e^{-\alpha t},0\le t\le \infty .$ The probability that its value lies between 100 and 200 hours is

1. $e^{-100\alpha }-e^{-200\alpha }$
2. $e^{-100}-e^{-200}$
3. $e^{-100\alpha }+e^{-200\alpha }$
4. $e^{-200\alpha }-e^{-100\alpha }$

Q. Probability density function of a random variable x is given below:

$f\left(x\right)=(\begin{array}{cc}0.25\mathrm{if}& 1\le x\le 5\\ 0& \mathrm{otherwise}\end{array}$

$P(x\le 4)$ is

1. $\frac{3}{4}$
2. $\frac{1}{2}$
3. $\frac{1}{4}$
4. $\frac{1}{8}$

Q.

Two random variables X and Y are distributed according to

$f_{x,y}(x,y)=(\begin{array}{cc}(x,y),& 0\le x\le 1,0\le y\le 1\\ 0,& \mathrm{otherwise}.\end{array}$

The probability $P(X+Y\le 1)\text{is}$ .

1. 0.33

Q. Assume that in a traffic junction, the cycle of the traffic signal lights is 2 minutes of green (vehicle does not stop) and 3minutes of red (vehicle stops). Consider that the arrival tme of vehicles at the junction is uniformly distributed over 5 minute cycle. The expected waiting time (in minutes) for the vehicle at the junction is

1. 0.9

Q.

Find the value of $\lambda$ such that the function f(x) is a valid probability density function.
$f\left(x\right)=\lambda (x-1)(2-x)\text{for}1\le x\le 2$
=0 otherwise

1. 6

Q. $P_x\left(X\right)=M{e}^{(-2|x|)}+N{e}^{(-3|x|)}$ is the probability density function for the real random variable X, over the entire x-axis, M and N are both positive real numbers. The equation relating M and N is

1. $M+\frac{2}{3}N=1$
2. $2M+\frac{1}{3}N=1$
3. M + N = 1
4. M + N = 3

Q. Consider two identically distributed zero-mean random variables U and V. Let the cumulative distribution functions of U and 2V be F(x) and G(x) respectively. Then, for all values of x

1. $F\left(x\right)-G\left(x\right)\le 0$
2. $F\left(x\right)-G\left(x\right)\ge 0$
3. $F\left(x\right)-G\left(x\right)x\le 0$
4. $F\left(x\right)-G\left(x\right).x\ge 0$

Q. The graph of a function f(x) is shown in the figure.

For f(x) to be a valid probability density function, the value of h is

1. 1/3
2. 2/3
3. 1
4. 3

Q. The cumulative distributino function of a random variable x is the probability that X takes the value

1. less than or equal to x
2. equal to x
3. greater than x
4. zero

Q.

Lifetime of an electric bulb is a random variable with density $f\left(x\right)=k{x}^2,$ where x is measured in years. If the minimum and maximum lifetimes of bulb are 1 and 2 years respectively, then the value of k is _____.

1. 0.428

Q.

The probability density function of evaporation E on any day during a year in watershed is given by $f\left(E\right)=(\begin{array}{cc}\frac{1}{5}& 0\le E\le 5\mathrm{mm}/\mathrm{day}\\ 0& \mathrm{otherwise}\end{array}$
The probability that E lies in between 2 and 4 mm/day in a day in watershed is (in decimal)

1. 0.4

Q.

Given that x is a random variable in range $[0,\infty ]$ with a probability density function $e^{-x/2}/K,$ the value of the constant K is ______ .

1. 2

Q.

A branch of a certain bank in New York City has six ATMs. Let $X$ represent the number of machines in use at a particular time of day. The cdf of $X$ is as follows:
$f\left(x\right)=\left\{\begin{array}{l}\begin{array}{c}0x<1\\ .601\le x<1\\ \begin{array}{c}.191\le x<2\\ .392\le x<3\\ \begin{array}{c}.673\le x<4\\ .924\le x<5\\ .975\le x<6\end{array}\end{array}\end{array}\\ 16\le x\end{array}\right.$
Calculate the following probabilities directly from the cdf:
$P\left(2\right),$that is, $P(X=2)$

Q.

Given that x is a random variable in range $[0,\infty ]$ with a probability density function $e^{-x/2}/K,$ the value of the constant K is ______ .

1. 2