Let n ≥ 3 . A list of numbers 0 < x 1 < x 2 < … < x n has mean μ and standard deviation σ . A new list of numbers is made as follows: y 1 = 0 , y 2 = x 2 , … . , y n - 1 = x n - 1 , y n = x 1 + x n . The mean and the standard deviation of the new list are μ ^ & σ ^ . Which of the following is necessarily true?

μ may or may not be equal to μ ^

EASY

Earn 100

The numbers of cars passing a fixed point on a road can be modelled as a random variable having a Poisson distribution with mean two cars per hours. The number of lorries passing the same point on the road can be modelled as a random variable having a Poisson distribution with mean three lorries per hour. Assuming that the random variables are independent, find the probability that in a randomly chosen hour a total of exactly four of these vehicles pass that point on the road.

Important Questions on Linear Combinations of Random Variables

HARD

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

The standard deviations of

x_{i} (i = 1, 2, \dots . 10)

and

y_{i} (i = 1, \dots 10)

are respectively

a

and

b

\bar{x}, \bar{y}

are the means of these two sets of observations respectively. If

z_{i} = (x_{i} - \bar{x}) (y_{i} - \bar{y})

and

\sum_{i = 1}^{10} z_{i} = c

, then the standard deviation of the observations

(x_{i} - y_{i}), (i = 1, 2, \dots 10)

EASY

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 set $(10, 10, 10, 10, 10)$ is

MEDIUM

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

For a group of

100

students, the mean

{\bar{x}}_{1}

and the standard deviation

σ_{1}

of their marks were found to be

40

and

15

respectively. Later it was observed that the scores

40

and

50

were misread as

30

and

60

respectively. If the mean and the standard deviation with the corrected observations of the scores are

{\bar{x}}_{2}

and

σ_{2}

respectively, then

MEDIUM

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

The S.D of

4, 5, 6, 7, \dots . ., 13

x

, then the S.D of

14, 15, \dots \dots 23

MEDIUM

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

In normal distribution what is the ratio of

QD : MD : SD

EASY

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

In a series of observations half of them equal to

a

and remaining half equal to

- a

. If the S.D. of the observations is

2

, then

| a | =

MEDIUM

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

If the mean and the standard deviation of the data

3, 5, 7, a, b

are

5

and

2

respectively, then

a

and

b

are the roots of the equation:

MEDIUM

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

Find

SD

1, 2, 3, 4, 5, 6, 7, 8, 9

MEDIUM

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

For a certain type of mobiles, the length of time between charges of the battery is normally distributed with a mean of

50

hours and a standard deviation of

15

hours. A person owns one of these mobiles and wants to know the probability that the length of time will be between

50

and

70

hours is (Given

ϕ (1.33) = 0.9082, ϕ (0) = 0.5

)

MEDIUM

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

Which of the following is not true regarding the normal distribution?

MEDIUM

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 numbers

31, 32, 33 \dots \dots . 46, 47

EASY

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

If we change the parameter (s) of a ________ distribution the Sharpe of probability curve does not change.

EASY

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

Standard deviation of a collection of data is $2 \sqrt{2}$ . If each value is multiplied by $3$ , then the standard deviation of the new data is:

EASY

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 a variable

x

10

. Then the standard deviation of

50 + 5 x

EASY

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

1

9

natural numbers is ____________

HARD

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

In a given data set, mean of

40

observations is

50

and standard deviation is

12

. Two readings which were

20 & 25

mistakenly taken as

40

and

45

. Find correct variance of data set.

EASY

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

- 1, 0, 1, k

\sqrt{5}

where

k > 0

, then

k

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

Let

n \geq 3

. A list of numbers

0 < x_{1} < x_{2} < \dots < x_{n}

has mean

μ

and standard deviation

σ

. A new list of numbers is made as follows:

y_{1} = 0, y_{2} = x_{2}, \dots ., y_{n - 1} = x_{n - 1}, y_{n} = x_{1} + x_{n}

. The mean and the standard deviation of the new list are

\hat{μ} & \hat{σ}

. Which of the following is necessarily true?

EASY

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

Area covered normal curve by

(μ \pm 3 σ)

is ______.

EASY

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

The quartile deviation of a normal distribution with mean

10

and standard deviation

4

is __________

Important Questions on Linear Combinations of Random Variables

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