Uniform Distribution

Q.

Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is .

1. 0.25

Q. The independent random variables X and Y are uniformly distributed in the interval [-1, 1]. The probability that max [X, Y] is less than 1/2 is

1. 3/4
2. 9/16
3. 1/4
4. 2/3

Q. Let the probability density function of a random variable, X, be given as:

$f_x\left(x\right)=\frac{3}{2}{e}^{-3x}u\left(x\right)+a{e}^{4x}u(-x)$

where u(x) is the unit step function.

Then the value of 'a' and Prob $X\le 0,$ respectively are

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

Q. A random variable is uniformly distributed over the interval 2 to 10. Its variance will be

1. 16/3
2. 6
3. 256/9
4. 36

Q. The probability density function $F\left(x\right)=a{e}^{-b|x|},$ where x is a random variable whose allowable value range is from $x=-\infty tox=+\infty .$ The CDF for this function for $x\ge 0$ is

1. $\frac{a}{b}{e}^{bx}$
2. $\frac{a}{b}(2-e^{-bx})$
3. $-\frac{a}{b}{e}^{bx}$
4. $-\frac{a}{b}(2+e^{-bx})$

Q. X is a uniformly distributed random variable that takes values between 0 and 1. The value of $E\left(X^3\right)$ will be

1. 0
2. 1/8
3. 1/4
4. 1/2

Q. If $x$ is uniformly distributed over (0, 15), the probability that $5<x<9$ is _____

1. $\frac{1}{3}$
2. $\frac{4}{15}$
3. $\frac{2}{5}$
4. $\frac{7}{15}$

Q. The standard deviation of a uniformly distributed random variable between 0 and 1 is

1. $\frac{1}{\sqrt{12}}$
2. $\frac{1}{\sqrt{3}}$
3. $\frac{5}{\sqrt{12}}$
4. $\frac{7}{\sqrt{12}}$

Q. A probability density function is of the form $p\left(x\right)=K{e}^{-\alpha \left|x\right|},xϵ(-\infty ,\infty ),$ the value of K is

1. 0.5
2. 1
3. $0.5\alpha$
4. $\alpha$

Q.

$\text{Let}{X}_1,X_2\text{and}{X}_3$ be independent and identically distributed random variables with the uniform distribution on [0, 1]. The probability $p\left\{x_1\text{is the largest}\right\}\text{is}$

1. 0.33

Q. For a random variable x having the PDF shown in the figure given below

the mean and the variance are, respectively

1. 0.5 and 0.66
2. 2.0 and 1.33
3. 1.0 and 0.66
4. 1.0 and 1.33

Q. The variable x takes a value between 0 and 10 with uniform probability distribution. The variable y takes a value between 0 and 20 with uniform probability distribution. The probability of the sum of variables (x + y) being greater than 20 is

1. 0.33
2. 0.25
3. 0
4. 0.50

Q. For the function $f\left(x\right)=a+bx,0\le x\le 1,$ to be a valid probability density function, which one of the following statements is correct?

1. a = 1, b = 4
2. a = 0.5, b = 1
3. a = 0, b = 1
4. a = 1, b = -1