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12th Business Maths Monthly Test - 3 ( Random Variable and Mathematical expectation )

St. Britto Hr. Sec. School - Madurai

12th Business Maths Monthly Test - 3 ( Random Variable and Mathematical expectation )-Aug 2020

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

A B

1. S.D of X a. \(\sigma { x }^{ 2 }\)

2. V(x) b. \(\overset { \infty }{ \underset { -\infty }{ \int } } \)x.f(x)dx

3. E(X²) - [E(X)]² c. + \(\sqrt { Var(x) } \)

4. E(X) d. V(X)

1-c, 2-d, 3-a, 4-b
1-d, 2-c, 3-b, 4-a
1-c, 2-a, 3-d, 4-b
1-a, 2-d, 3-b, 4-c

The following table is the probability distribution of random variable X

X 1 2 3 4 5 6

P(X) 0.1 2k k 0.2 3k 0.1

The value of k is _____________.

0.2
0.1
0.3
0.4

Consider a random variable. X with probability density function

f(x) = \(\begin{cases} 4{ x }^{ 3 },\quad if\quad 0<x<1 \\ 0,\quad \quad otherwise \end{cases}\)

A B

1. E(X) a. \(\frac { 2 }{ 75 }\)

2. E(X²) b. \(\frac { 4 }{ 5 }\)

3. V(X) c. \(\frac { 4 }{6 }\)

4. V(4) d. 0

The correct match is:

1-a, 2-c, 3-d, 4-b
1-d, 2-a, 3-c, 4-b
1-b, 2-c, 3-a, 4-d
1-c, 2-d, 3-a, 4-b

The probability density function p(x) cannot exceed

zero
one
mean
infinity

If X is a continuous random variable., then which of the following is incorrect?

F'(x)=f(x)
F(\(\infty \))= 1, F(-\(\infty \)) = 0
P(a \(\le\) X \(\le\) b) = F(b) - F(a)
P(a \(\le\) X < b) \(\neq \) F(b) - F(a)

The distribution function F(x) is equal to

P(X-x)
P(X\(\le\)x)
P(X\(\ge\)x)
all of these

Given E(X) = 5 and E(Y) = -2, then E(X-Y) is

3
5
7
-2

Variance of the random variable. X is 4. Its mean is 2. Then E(X²) is

2
4
6
8

A random variable X has the following probability distribution

X 0 1 2 3 4 5

P(X = x) \(\frac { 1 }{ 4 }\) 2a 3a 4a 5a \(\frac { 1 }{ 4 }\)

Then P(1\(\le\)x\(\le\)4) is

\(\frac { 10 }{ 21 } \)
\(\quad \\ \frac { 2 }{ 7 } \)
\(\frac { 1 }{ 14 } \)
\(\\ \frac { 1 }{ 2 } \)

A discrete probability distribution may be represented by

table
graph
mathematical equation
all of these

If X is a discrete random variable. then P(X\(\ge\)a)= _______________.

P(X < a)
1-P(X \(\le\) a)
1-P(X < a)
0

A discrete probability function p(x) is always

non-negative
negative
one
zero

If X is a discrete random variable and p x ( ) is the probability of X , then the expected value of this random variable is equal to

\(\sum { f(x) } \)
\(\sum { [x+f(x)] } \)
\(\sum { f(x)+x } \)
\(\sum { xp(x) } \)

If we have f(x)= 2x, 0 \(\le\) x \(\le\) 1, then f(x) is a

probability distribution
probability density function
distribution function
continuus random variable

The random variable. X has variance 4 and E(X²) = 8. Then the mean of X is

2\(\sqrt 3\)
4
2
\(\sqrt 2\)

A discrete probability function p(x) is always non-negative and always lies between

0 and \(\infty \)
0 and 1
-1 and +1
-\(\infty \) and +\(\infty \)

A formula or equation used to represent the probability distribution of a continuous random variable is called

probability distribution
distribution function
probability density function
mathematical expectation

If E(X) = \(\frac { 1 }{ 2 }\), E(X²) = \(\frac { 1 }{ 4 }\), then V(X) is

0
\(\frac { 1 }{ 4 }\)
\(\frac { 1 }{ 2 }\)
1

A discrete random variable. X has the probability mass function p(x), then _______ is true.

0 \(\le\) P(X) \(\le\) 1
P(X) \(\ge\) 0
P(X) \(\le\) 1
0 < P (X) < 1

In the following distribution function

X 1 2 3 4 5 6

F(X) \(\frac { 1 }{ 6 }\) \(\frac { 2 }{ 6 }\) \(\frac { 3 }{ 6 }\) \(\frac { 4 }{ 6 }\) \(\frac { 5 }{ 6 }\) 1

A B

1. P(X=1) a. \(\frac { 1 }{ 6 }\)

2. P(X \(\le\) 2) b. \(\frac { 3 }{ 6 }\)

3. P(X=5) c. \(\frac { 2 }{ 6 }\)

4. P(X \(\ge\) 4) d. \(\frac { 1 }{ 6 }\)

The correct match is

1-a, 2-b, 3-c, 4-d
1-b, 2-c, 3-d, 4-a
1-c, 2-d, 3-a, 4-b
1-a, 2-c, 3-d, 4-b

Part - B

Generic 5 - Answer All Question

If f (x) is defined by f(x)=k2-2x, 0\(\le\)x<\(\infty\)is a density function. Determine the constant k and also find mean.

Determine the mean and variance of a discrete random variable, given its distribution as follows.

X=x 1 2 3 4 5 6

F_x(x) \(\frac{1}{6}\) \(\frac{2}{6}\) \(\frac{3}{6}\) \(\frac{4}{6}\) \(\frac{5}{6}\) 1

A commuter train arrives punctually at a station every 25 minutes. Each morning, a commuter leaves his house and casually walks to the train station. Let X denote the amount of time, in minutes, that commuter waits for the train from the time he reaches the train station. It is known that the probability density function of X is
f(x)={\(\frac { 1 }{ 25 } ,\quad for0<x<25\\ 0,\quad otherwise,\)
Obtain and interpret the expected value of the random variable X.

An urn contains four balls of red, black, green and blue colours. There is an equal probability of getting any coloured ball. What is the expected value of getting a blue ball out of 30 experiments with replacement?

Suppose the probability mass function of the discrete random variable is

X=x 0 1 2 3

p(x) 0.2 0.1 0.4 0.3

What is the value of E(3X + 2X²) ?

The probability distribution of a random variable X is

X 1 2 4 2A 3A 5A

P(X) \(\frac { 1 }{ 2 } \) \(\frac15\) \(\frac{3}{25}\) \(\frac { 1 }{ 10 } \) \(\frac { 1 }{ 25 } \) \(\frac { 1 }{ 25 } \)

Calculate (i) A if E(X) = 2.94 (ii) V(X)

The probability density function of a random variable X is
f(x)=ke^-|x|, -∞<x< ∞
Find the value of k and also find mean and variance for the random variable.

Suppose the life in hours of a radio tube has the probability density function
f(x)={\({ e }^{ \frac { x }{ 100 } },when\quad x\ge 100\\ 0,\quad when\quad x<100\)
Find the mean of the life of a radio tube.

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