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The continuous random variable $X$ represents the amount of sunshine in hours between noon and $4$ pm at a skiing resort in the high season. The probability density function, $f (x)$ , of $X$ is modelled by $f (x) = \{\begin{cases} \frac{3 x^{2}}{64} for 0 ⩽ x ⩽ 4 \\ 0 otherwise \end{cases}$ . Find $P (2 < x < 4)$ .

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Important Questions on Continuous Random Variables

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Continuous Random Variables>Probability Distribution of Continuous Random 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) =

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Continuous Random Variables>Probability Distribution of Continuous Random Variables

For the following probability distribution:

$X$	$1$	$2$	$3$	$4$
$P (X)$	$\frac{1}{10}$	$\frac{1}{5}$	$\frac{3}{10}$	$\frac{2}{5}$

$E (X^{2})$ is equal to

HARD

Mathematics (9709)>P-6: Statistics and Probability 2>Continuous Random Variables>Probability Distribution of Continuous Random 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>Continuous Random Variables>Probability Distribution of Continuous Random Variables

Given below is the probability distribution of discrete random variable $X$

$X = x$	$1$	$2$	$3$	$4$	$5$	$6$
$P [X = x]$	$K$	$0$	$2 K$	$5 K$	$K$	$3 K$

Then, $P [X \geq 4] =$

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Continuous Random Variables>Probability Distribution of Continuous Random 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>Continuous Random Variables>Probability Distribution of Continuous Random 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>Continuous Random Variables>Probability Distribution of Continuous Random 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:

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Continuous Random Variables>Probability Distribution of Continuous Random Variables

What is the mean of $f (x) = 3 x + 2$ where $x$ is a random variable with probability distribution

$X = x$	$1$	$2$	$3$	$4$
$P (X = x)$	$\frac{1}{6}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{6}$

EASY

Mathematics (9709)>P-6: Statistics and Probability 2>Continuous Random Variables>Probability Distribution of Continuous Random 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>Continuous Random Variables>Probability Distribution of Continuous Random 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>Continuous Random Variables>Probability Distribution of Continuous Random Variables

For the probability distribution given by

$X = x_{i}$	$0$	$1$	$2$
$P_{i}$	$25 / 36$	$5 / 18$	$1 / 36$

The standard deviation $(σ)$ is

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Continuous Random Variables>Probability Distribution of Continuous Random 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 =

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Continuous Random Variables>Probability Distribution of Continuous Random Variables

The probability distribution of a discrete random variable $X$ is given in the following table:

$X = x$	$0$	$1$	$2$
$P (x)$	$4 C^{3}$	$4 C - 13 C^{2}$	$7 C - 1$

; $C > 0$ then $C =$ ________.

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Continuous Random Variables>Probability Distribution of Continuous Random 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>Continuous Random Variables>Probability Distribution of Continuous Random Variables

The mean and standard deviation of random variable

X

are

10

and

5

respectively, then

E {(\frac{X - 15}{5})}^{2} =

_________.

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Continuous Random Variables>Probability Distribution of Continuous Random Variables

Let $X$ be a random variable with distribution.

$x$	$- 2$	$- 1$	$3$	$4$	$6$
$P (X = x)$	$\frac{1}{5}$	$a$	$\frac{1}{3}$	$\frac{1}{5}$	$b$

If the mean of $X$ is $2.3$ and variance of $X$ is $σ^{2},$ then $100 σ^{2}$ is equal to :

HARD

Mathematics (9709)>P-6: Statistics and Probability 2>Continuous Random Variables>Probability Distribution of Continuous Random Variables

Let

X

be a random variable such that the probability function of a distribution is given by

P (X = 0) = \frac{1}{2}, P (X = j) = \frac{1}{3^{j}} (j = 1, 2, 3, \dots, \infty) .

Then the mean of the distribution and

P (X

is positive and even

)

respectively, are:

HARD

Mathematics (9709)>P-6: Statistics and Probability 2>Continuous Random Variables>Probability Distribution of Continuous Random Variables

If '

X

' has a binomial distribution with parameters

n = 6, p

and

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

, then

p =

MEDIUM

Mathematics (9709)>P-6: Statistics and Probability 2>Continuous Random Variables>Probability Distribution of Continuous Random 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>Continuous Random Variables>Probability Distribution of Continuous Random 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 :

The continuous random variable X represents the amount of sunshine in hours between noon and 4 pm at a skiing resort in the high season. The probability density function, f(x), of X is modelled by f(x)=3x264 for 0⩽x⩽40 otherwise . Find P(2<x<4).

Important Questions on Continuous Random Variables

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