Mathematical Expectation

Q. The random variable X takes on the values 1, 2 (or) 3 with probabilities

$\frac{2+5P}{5},\frac{1+3P}{5}\text{and}\frac{1.5+2P}{5}$

respectively the values of P and E(X) are respectively

1. 0.05, 1.87
2. 1.90, 5.87
3. 0.05, 1.10
4. 0.25, 1.40

Q.

If a random variable X has a Poisson distribution with mean 5, then the expectation

$E\left[\right(X+2{)}^2]\text{equals}$.

1. 54

Q. If E denotes expectation, the variance of a random variable X is given by

1. $E\left(X^2\right)-E^2\left(X\right)$
2. $E\left(X^2\right)+E^2\left(X\right)$
3. $E\left(X^2\right)$
4. $E^2\left(X\right)$

Q.

A six - face fair dice is rolled a large number of times. The mean value of the outcomes is .

1. 3.5

Q.

The probability density function of a random variable X is $P_x\left(x\right)=e^{-x}\text{for}x\ge 0\text{and}0\text{otherwise.}$ The expected value of the function $g_x\left(x\right)=e^{3x/4}\text{is}$ .

1. 4

Q.

A fair die with faces {1, 2, 3, 4, 5, 6} is thrown repeatedly till '3' is observed for the first time. Let X denote the number of times the die is thrown. The expected value of X is .

1. 6

Q. Let X and Y be two independent random variables. Which one of the relations between expectation (E), variance (Var) and covariance (Cov) given below is FALSE?

1. E(XY) = E(X) E(Y)
2. Cov (X, Y) = 0
3. Var (X + Y) = Var (X) + Var (Y)
4. $E\left(X^2{Y}^2\right)=\left(E\right(X\left){)}^2\right(E\left(Y\right){)}^2$

Q. An exam paper has 150 multiple choice questions of 1 mark each, with each question having four choices. Each incorrect answer fetches -0.25 marks. Suppose 1000 students choose all their answer randomly with uniform probability. The sum total of the expected marks obtained by all the students is

1. 0
2. 2550
3. 7525
4. 9375

Q. A random variable X has a probability density function
$f\left(x\right)=(\begin{array}{cc}{\mathrm{ke}}^n{e}^{-x};& x\ge 0\\ 0;& \mathrm{otherwise}\end{array}$ (n is an integer)
with mean 3. The values of {k, n} are

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

Q. Among the four normal distribution with probability density function as shown below, which one has the least density the lowest variance?

1. I
2. II
3. III
4. IV

Q.

Let the random variable X represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of X is .

1. 1.5

Q. In the following table, x is a discrete random variable and p(x) is the probability density function. The standard deviation of x is

x	1	2	3
p(x)	0.3	0.6	0.1

1. 0.18
2. 0.36
3. 0.54
4. 0.6

Q.

The mode of the Binomial distribution for which mean and standard deviation are $10$ and $\sqrt{5}$ respectively, is

1. $7$

2. $8$

3. $9$

4. $10$

Q.

Passengers try repeatedly to get a seat reservaton in any train running between two stations until they are successful. If there is 40% chance of getting reservation in any attempt by a passenger then the average number of attempts that passengers need to make to get a seat reserved is

1. 3

Q.

The probability density function of a random variables x is
$\begin{array}{cc}f\left(x\right)=\frac{x}{4}(4-x^2);& \mathrm{for}0\le x\le 2\\ =0;& \mathrm{otherwise}\end{array}$
The mean ${\mu }_x$ of the random variable is

1. 16.15

Q.

Find the number of subsets of the set $1,2,3,4,5,6,7,8,9,10,11$ having $4$ elements.

Q. For a single server with Poisson arrival and exponential service time, the arrival rate is 12 per hour. Which one of the following service rates will provide a steady state finite queue length?

1. 6 per hour
2. 10 per hour
3. 12 per hour
4. 24 per hour

Q.

Consider the following probability mass function (p.m.f.) of a random variable X.

$p(x,q)=(\begin{array}{cc}q& \mathrm{if}X=0\\ 1-q& \mathrm{if}X=1\\ 0& \mathrm{otherwise}\end{array}$

If q = 0.4, the variance of X is .

1. 0.24

Q.

The probability distribution of a random variable $X$ is given as follows. The value of $p$ is

$X$	$-5$	$-4$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$	$5$
$p(X=x)$	$p$	$2p$	$3p$	$4p$	$5p$	$7p$	$8p$	$9p$	$10p$	$11p$	$12p$

The value of $p$is

1. $\frac{1}{72}$

2. $\frac{3}{73}$

3. $\frac{5}{72}$

4. $\frac{1}{74}$

5. $\frac{1}{73}$

Q. A machine produces 0, 1 or 2 defective pieces in a day with associated probability of 1/6, 2/3 and 1/6, respectively. The mean value and the variance of the number of defective pieces produced by the machine in a day, respectively, are

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

Q.

The variance of the random variable X with probability density function $f\left(x\right)=\frac{1}{2}|x|e^{-|x|}$ is .

1. 6

Q.

Each of the nine words in the sentence, "The Quick brown fox jumps over the lazy dog" is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is .
(The answer should be rounded to one decimal place)

1. 3.88

Q. The following data about the flow of liquid was observed in a continuous chemical process plant.

Flow rate (litres/sec)	Frequency
7.5 to 7.7	1
7.7 to 7.9	5
7.9 to 8.1	35
8.1 to 8.3	17
8.3 to 8.5	12
8.5 to 8.7	10

Mean flow rate of the liquid is

1. 8.00 litres/sec
2. 8.06 litres/sec
3. 8.16 litres/sec
4. 8.26 litres/sec

Q.

Let X be a random variable which is uniformly chosen from the set of positive odd numbers less than 100. The expectation, E[x], is

1. 50

Q.

The probability density function on the interval [a, 1] is given by $1/x^2$ and outside this interval the value of the function is zero. The value of a is

1. 0.5

Q. A discrete random variable X takes value from 1 to 5 with probabilities as shown in the table. A student calculate the mean of X as 3.5 and her teacher calculates the variance to X as 1.5 which of the following statement is true?

K	1	2	3	4	5
P(X = K)	0.1	0.2	0.4	0.2	0.1

1. Both the student and the teacher are right
2. Both the student and the teacher are wrong
3. The student is wrong but the teacher is right
4. The student is right but the teacher is wrong

Q. A person decides to toss a fair coin repeatedly until he gets a head. He will make at most 3 tosses. Let the random variable Y denote the number of heads. The value of var(Y). where var(.) denotes the variance, equals:

1. $\frac{7}{8}$
2. $\frac{49}{64}$
3. $\frac{7}{64}$
4. $\frac{105}{64}$

Q.

Suppose that $x$ is the number of cars per minute that pass through a certain intersection, and that $x$ has the Poisson distribution. If the mean of $x$ is equal to $9\text{cars}$, what is the variance of $x$?

1. $9^2\text{cars}$

2. $9{\text{cars}}^2$

Q.

Consider the random process

X(t) = U + Vt.

where U is a zero-mean Gaussian random variable and V is a random variable uniformly distributed between 0 and 2. Assume that U and V are statistically independent. The mean value of the random process at t = 2 is

1. 2

Q. If f(x) and g(x) are two probability density functions,

$f\left(x\right)=(\begin{array}{cc}\frac{x}{a}+1:& -a\le x<0\\ -\frac{x}{a}+1:& 0\le x\le a\end{array}$

$g\left(x\right)=(\begin{array}{cc}-\frac{x}{a}:& -a\le x<0\\ \frac{x}{a}:& 0\le x\le a\\ 0:& \mathrm{otherwise}\end{array}$

Which of the following statements is true?

1. Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are same.
2. Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are different.
3. Mean of f(x) and g(x) are different; Variance of f(x) and g(x) are same.
4. Mean of f(x) and g(x) are different; Variance of f(x) and g(x) are different.