EASY

Earn 100

The mean and variance of the random variable $X$ are $5.8$ and $3.1$ , respectively. The random variable $S$ is the sum of three independent values of $X$ . The independent random variable $T$ is defined by $T = 3 X + 2$ .

Find the mean and variance of $S - T$ .

Important Questions on Linear Combinations of Random Variables

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

The variance of first

20

natural numbers is

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

A data consists of

E (a X + b) = a \times E (X) + b

observations:

x_{1}, x_{2}, \dots, x_{n} .

\sum_{i = 1}^{n} {(x_{i} + 1)}^{2} = 9 n

and

\sum_{i = 1}^{n} {(x_{i} - 1)}^{2} = 5 n

, then the standard deviation of this data is

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

A student scores the following marks in five tests:

45,54,41,57,43

. His score is not known for the sixth test. If the mean score is

48

in the six tests, then the standard deviation of the marks in six tests is

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of

10

to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

Let $\bar{x}$ , $M$ and $σ^{2}$ be respectively the mean, mode and variance of $n$ observations $x_{1}, x_{2}, . ..., x_{n}$ and $d_{i} = - x_{i} - a, i = 1, 2, . ..., n,$ where $a$ is any number.

Statement I: Variance of $d_{1}, d_{2}, . . ., d_{n}$ is $σ^{2}$ .

Statement II: Mean and mode of $d_{1}, d_{2}, . . . ., d_{n}$ are $- \bar{x} - a$ and $- M - a$ , respectively.

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

The mean and variance for seven observations are

8

and

16

respectively. If

5

of the observations are

2, 4, 10, 12, 14,

then the product of the remaining two observations is

HARD

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

If the variance of the first

n

natural numbers is

10

and the variance of the first

m

even natural numbers is

16,

then the value of

m + n

is equal to

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

The mean and variance of

20

observations are found to be

10

and

4,

respectively. On rechecking, it was found that an observation

9

was incorrect and the correct observation was

11,

then the correct variance is

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

\sum_{i = 1}^{9} (x_{i} - 5) = 9

and

\sum_{i = 1}^{9} {(x_{i} - 5)}^{2} = 45

, then the standard deviation of the

9

items

x_{1}, x_{2}, \dots ., x_{9}

HARD

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

The mean of

5

observations is

5

and their variance is

12.4

. If three of the observations are

1, 2 & 6

; then the value of the remaining two is :

HARD

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

If the data

x_{1}, x_{2}, \dots x_{10}

is such that the mean of first four of these is

11,

the mean of the remaining six is

16

and the sum of squares of all of these is

2000

, then the standard deviation of this data is:

HARD

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

The standard deviation of the data

6, 5, 9, 13, 12, 8, 10

HARD

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

Let the observation

x_{i} (1 \leq i \leq 10)

satisfy the equations

\sum_{i = 1}^{10} (x_{i} - 5) = 10

\sum_{i = 1}^{10} {(x_{i} - 5)}^{2} = 40

. If

μ

and

λ

are the mean and the variance of the observations,

x_{1} - 3, x_{2} - 3, . . . ., x_{10} - 3,

then the ordered pair

(μ, λ)

is equal to

HARD

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

The mean and the variance of five observations are

4

and

5.20

, respectively. If three of the observations are

3, 4

and

4

; then the absolute value of the difference of the other two observations, is :

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

The mean and the standard deviation (s.d.) of

10

observations are

20

and

2

respectively. Each of these

10

observations is multiplied by

p

and then reduced by

q,

where

p \neq 0

and

q \neq 0 .

If the new mean and new s.d. become half of their original values, then

q

is equal to

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

The mean and variance of

8

observations are

10

and

13.5,

respectively. If

6

of these observations are

5, 7, 10, 12, 14, 15,

then the absolute difference of the remaining two observations is :

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

If the variance of the terms in an increasing

A . P .

b_{1} b_{2}, b_{3}, \dots \dots . ., b_{11}

90

, then the common difference of this

A . P .

HARD

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

If the standard deviation of the numbers

2, 3, a

and

11

3.5

, then which of the following is true ?

HARD

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

The sum of

100

observations and the sum of their squares are

400 & 2475

, respectively. Later on, three observations

3, 4 & 5

were found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations is

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Linear Combinations of Random Variables>Mean and Variance of Linear Combination of Variables

The variance of

20

observations is

5 .

If each observation is multiplied by

3,

then what is the new variance of the resulting observations?

The mean and variance of the random variable X are 5.8 and 3.1, respectively. The random variable S is the sum of three independent values of X. The independent random variable T is defined by T=3X+2. Find the mean and variance of S-T.

Important Questions on Linear Combinations of Random Variables

Learn and Prepare for any exam you want !

The mean and variance of the random variable $X$ are $5.8$ and $3.1$ , respectively. The random variable $S$ is the sum of three independent values of $X$ . The independent random variable $T$ is defined by $T = 3 X + 2$ .

Find the mean and variance of $S - T$ .