The probability distribution table of random variable $X$ is shown below. Find the variance of random variable $Y = 10 X + 5$. [3 points]

$X$	0	1	2	3	Total
$\mathrm { P } ( X )$	$\frac { 2 } { 10 }$	$\frac { 3 } { 10 }$	$\frac { 3 } { 10 }$	$\frac { 2 } { 10 }$	1

The following is a probability distribution table for the random variable $X$.

$X$	$k$	$2 k$	$4 k$	Total
$\mathrm { P } ( X = x )$	$\frac { 4 } { 7 }$	$a$	$b$	1

If $\frac { 4 } { 7 } , a , b$ form a geometric sequence in this order and the mean of $X$ is 24, find the value of $k$. [3 points]

(Probability and Statistics) A coin is tossed three times, and based on the results, a score is obtained as a random variable $X$ according to the following rules. (가) If the same face does not appear consecutively, the score is 0 points. (나) If the same face appears consecutively exactly twice, the score is 1 point. (다) If the same face appears consecutively three times, the score is 3 points.
What is the variance $\mathrm{V}(X)$ of the random variable $X$? [3 points]
(1) $\frac{9}{8}$
(2) $\frac{19}{16}$
(3) $\frac{5}{4}$
(4) $\frac{21}{16}$
(5) $\frac{11}{8}$

The probability distribution table of the random variable $X$ is as follows.

$X$	0	1	2	Total
$\mathrm { P } ( X = x )$	$\frac { 2 } { 7 }$	$\frac { 3 } { 7 }$	$\frac { 2 } { 7 }$	1

What is the value of the variance $\mathrm { V } ( 7 X )$ of the random variable $7 X$? [3 points]
(1) 14
(2) 21
(3) 28
(4) 35
(5) 42

The probability distribution table of the random variable $X$ is as follows.

$X$	- 1	0	1	2	Total
$\mathrm { P } ( X = x )$	$\frac { 3 - a } { 8 }$	$\frac { 1 } { 8 }$	$\frac { 3 + a } { 8 }$	$\frac { 1 } { 8 }$	1

When $\mathrm { P } ( 0 \leqq X \leqq 2 ) = \frac { 7 } { 8 }$, what is the value of the expected value $\mathrm { E } ( X )$ of the random variable $X$? [3 points]
(1) $\frac { 1 } { 4 }$
(2) $\frac { 3 } { 8 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 5 } { 8 }$
(5) $\frac { 3 } { 4 }$

The probability mass function of a discrete random variable $X$ is $$\mathrm { P } ( X = x ) = \frac { a x + 2 } { 10 } ( x = - 1,0,1,2 )$$ What is the value of the variance $\mathrm { V } ( 3 X + 2 )$ of the random variable $3 X + 2$? (where $a$ is a constant.) [3 points]
(1) 9
(2) 18
(3) 27
(4) 36
(5) 45

The probability distribution of a random variable $X$ is shown in the table below.

$X$	0	1	2	Total
$\mathrm { P } ( X = x )$	$\frac { 1 } { 4 }$	$a$	$2a$	1

What is the value of $\mathrm { E } ( 4X + 10 )$? [3 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15

The probability distribution of a discrete random variable $X$ is shown in the table below.

$X$	- 5	0	5	Total
$\mathrm { P } ( X = x )$	$\frac { 1 } { 5 }$	$\frac { 1 } { 5 }$	$\frac { 3 } { 5 }$	1

Find the value of $\mathrm { E } ( 4 X + 3 )$. [3 points]

The probability distribution of the random variable $X$ is shown in the table below.

$X$	0.121	0.221	0.321	Total
$\mathrm { P } ( X = x )$	$a$	$b$	$\frac { 2 } { 3 }$	1

The following is the process of finding $\mathrm { V } ( X )$ when $\mathrm { E } ( X ) = 0.271$. Let $Y = 10 X - 2.21$. The probability distribution of the random variable $Y$ is shown in the table below.

$Y$	$-1$	0	1	Total
$\mathrm { P } ( Y = y )$	$a$	$b$	$\frac { 2 } { 3 }$	1

Since $\mathrm { E } ( Y ) = 10 \mathrm { E } ( X ) - 2.21 = 0.5$, $a =$ (가), $b =$ (나) and $\mathrm { V } ( Y ) = \frac { 7 } { 12 }$. On the other hand, since $Y = 10 X - 2.21$, we have $\mathrm { V } ( Y ) =$ (다) $\times \mathrm { V } ( X )$. Therefore, $\mathrm { V } ( X ) = \frac { 1 } { \text{(다)} } \times \frac { 7 } { 12 }$. When the values in (가), (나), and (다) are $p$, $q$, and $r$ respectively, find the value of $pqr$. (Here, $a$ and $b$ are constants.) [4 points]
(1) $\frac { 13 } { 9 }$
(2) $\frac { 16 } { 9 }$
(3) $\frac { 19 } { 9 }$
(4) $\frac { 22 } { 9 }$
(5) $\frac { 25 } { 9 }$

There is a bag containing 5 cards with the numbers $1, 3, 5, 7, 9$ written on them, one number per card. A trial is performed by randomly drawing one card from the bag, confirming the number on the card, and putting it back. This trial is repeated 3 times, and let $\bar{X}$ be the average of the three numbers confirmed. When $\mathrm{V}(a\bar{X} + 6) = 24$, what is the value of the positive number $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

A discrete random variable $X$ takes values that are integers from 0 to 4, and $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { | 2 x - 1 | } { 12 } & ( x = 0,1,2,3 ) \\ a & ( x = 4 ) \end{array} \right.$$ What is the value of $\mathrm { V } \left( \frac { 1 } { a } X \right)$? (Here, $a$ is a nonzero constant.) [3 points]
(1) 36
(2) 39
(3) 42
(4) 45
(5) 48

19. (12 points) The one-way driving time T between a school's new and old campuses depends only on road conditions. A sample of 100 observations was collected with the following results:

T (minutes)

25

30

35

40

Frequency

20

30

40

10

(I) Find the probability distribution of T and the mathematical expectation ET; (II) Professor Liu drives from the old campus to the new campus for a 50-minute lecture, then immediately returns to the old campus. Find the probability that the total time from leaving the old campus to returning is no more than 120 minutes.

18.
(1)
The probability distribution of $X$ is:
$$\begin{gathered} P ( X = 0 ) = 0.2 \\ P ( X = 20 ) = 0.8 \times ( 1 - 0.6 ) = 0.32 \\ P ( X = 100 ) = 0.8 \times 0.6 = 0.48 \end{gathered}$$
(2)
Given that type A questions are answered first, the mathematical expectation is...

Calculate the expectation and the variance of a variable following the distribution $\mathcal { R }$, where $\mathcal{R}$ is defined by $X ( \Omega ) = \{ - 1,1 \}$, $\mathbb { P } ( X = - 1 ) = \mathbb { P } ( X = 1 ) = \frac { 1 } { 2 }$.

Let $X$ be a random variable, and let $P ( X = x )$ denote the probability that $X$ takes the value $x$. Suppose that the points $( x , P ( X = x ) ) , x = 0,1,2,3,4$, lie on a fixed straight line in the $x y$-plane, and $P ( X = x ) = 0$ for all $x \in \mathbb { R } - \{ 0,1,2,3,4 \}$. If the mean of $X$ is $\frac { 5 } { 2 }$, and the variance of $X$ is $\alpha$, then the value of $24 \alpha$ is $\_\_\_\_$ .

The probability distribution of random variable $X$ is given by:

$X$	1	2	3	4	5
$P ( X )$	$K$	$2 K$	$2 K$	$3 K$	$K$

Let $p = P ( 1 < X < 4 \mid X < 3 )$. If $5 p = \lambda K$, then $\lambda$ is equal to

If the mean of the following probability distribution of a random variable $X$ :

X	0	2	4	6	8
$\mathrm { P } ( \mathrm { X } )$	$a$	$2a$	$a + b$	$2b$	$3b$

is $\frac { 46 } { 9 }$, then the variance of the distribution is
(1) $\frac { 173 } { 27 }$
(2) $\frac { 566 } { 81 }$
(3) $\frac { 151 } { 27 }$
(4) $\frac { 581 } { 81 }$

Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $x$ denote the number of defective oranges, then the variance of $x$ is
(1) $28/75$
(2) $18/25$
(3) $26/75$
(4) $14/25$

If the possible values of random variable $X$ are $1 , 2 , 3 , 4$ , and the probability $P ( X = k )$ is proportional to $\frac { 1 } { k }$ , then the probability $P ( X = 3 )$ is $\frac { \text{(9)(11)} } { \text{(10)(11)} }$ . (Reduce to lowest terms)