EASY

Earn 100

The number of accidents on a certain road has a Poisson distribution with mean $3.1$ per $12$ -week peroid

Given that the mean number of accidents per week is now $0.1$ , find the probability of a Type $II$ error.

Important Questions on Linear Combinations of Random Variables

MEDIUM

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

An unbiased coin is tossed

5

times. Suppose that a variable

X

is assigned the value

k

when

k

consecutive heads are obtained for

k = 3, 4, 5,

otherwise

X

takes the value

- 1 .

Then the expected value of

X,

HARD

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

In a game, a man wins Rs.

100

if he gets

5

6

on a throw of a fair die and loses Rs.

50

for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :

MEDIUM

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

A boy tosses fair coin

3

times. If he gets ₹

2 x

for

x

heads then his expected gain equals to ₹........

HARD

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

If '

X

' has a binomial distribution with parameters

n = 6, p

and

P (X = 2) = 12, P (X = 3) = 5

, then

p =

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 :

MEDIUM

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

A random variable $X$ has the following probability distribution

$X$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$P (X)$	$K - 1$	$3 K$	$K$	$3 K$	$3 K^{2}$	$K^{2}$	$K^{2} + K$

MEDIUM

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

A box contains

6

pens,

2

of which are defective. Two pens are taken randomly from the box. If random variable

x :

Number of defective pens obtained, then standard deviation of

x =

HARD

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

If the function

f

defined by

f (x) = \{\begin{cases} K (x - x^{2}); & if 0 < x < 1 \\ 0; & otherwise \end{cases}

is the probability density function of a random variable

X

, then the value of

P (X < \frac{1}{2})

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

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

Let

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

be ten observation of a random variable

X

. If

\sum_{i = 1}^{10} (x_{i} - p) = 3

and

\sum_{i = 1}^{10} {(x_{i} - p)}^{2} = 9

where

0 \neq p \in R

, then the standard deviation of these observations is:

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 :

HARD

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

Let

X

be a random variable which takes values

k

with the probability

k p,

where

k = 1, 2, 3, 4

and

p \in (0, 1),

then the standard deviation of

X

EASY

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

A random variable

X

has the following probability distribution:

\begin{matrix} \begin{matrix} X : & 1 & \begin{matrix} 2 & 3 & \begin{matrix} 4 & 5 \end{matrix} \end{matrix} \end{matrix} \\ P (X) : \begin{matrix} k^{2} & \begin{matrix} 2 k & k & \begin{matrix} 2 k & 5 k^{2} \end{matrix} \end{matrix} \end{matrix} \end{matrix}

Then,

P (X > 2)

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

A random variable

X

takes the values

0, 1, 2

. Its mean is

1.2

P (X = 0) = 0.3

, then

P (X = 1) =

HARD

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

Suppose

A

3 \times 3

matrix consisting of integer entries that are chosen at random from the set

\{- 1000, - 999, \dots 999, 1000\} .

Let

P

be the probability that either

A^{2} = - I

A

is diagonal, where

I

is the

3 \times 3

identity matrix. Then,

MEDIUM

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

A random variable $X$ has the following probability distribution:

$X = x_{i}$	$- 2$	$- 1$	$0$	$1$	$2$
$P (X = x_{i})$	$\frac{1}{6}$	$k$	$\frac{1}{4}$	$k$	$\frac{1}{6}$

The variance of this random variable 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

10

natural numbers

1, 1, 1, \dots, 1, k

is less than

10,

then the maximum possible integral value of

k

is ___________.

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

MEDIUM

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

The mean of five observations is

5

and their variance is

9.20 .

If three of the given five observations are

1, 3

and

8,

then a ratio of other two observations is

The number of accidents on a certain road has a Poisson distribution with mean 3.1 per 12-week peroid Given that the mean number of accidents per week is now 0.1, find the probability of a Type II error.

Important Questions on Linear Combinations of Random Variables

Learn and Prepare for any exam you want !

The number of accidents on a certain road has a Poisson distribution with mean $3.1$ per $12$ -week peroid

Given that the mean number of accidents per week is now $0.1$ , find the probability of a Type $II$ error.