Exercise 1

A pair of a die is thrown. The random variable, X, is defined as the sum of the obtained scores. Determine the probability distribution, the expected value, and variance.

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Exercise 2

A player throws a die. If a prime number is obtained, he gains to win an amount equal to the number rolled times 100 dollars, but if a prime number is not obtained, he loses an amount equal to the number rolled times 100 dollars. Calculate the probability distribution and the expected value of the described game.

Exercise 3

The first prize for a raffle is 5,000 dollars (with a probability of 0.001) and the second prize is 2,000 dollars (with a probability of 0.003). What is a fair price to pay for a single ticket in this raffle?

Exercise 4

Let X be a discrete random variable whose probability distribution is as follows:

x { p }_{ i }
0 0,1
1 0,2
2 0,1
3 0,4
4 0,1
5 0,1

1. Calculate the distribution function.

2. Calculate the following probabilities:

p (x < 4.5)

p (x \ge  3)

p (3 \le x < 4.5)

Exercise 5

A player tosses two coins into the air. He wins 1 dollar for the number of heads he will get. However, he will lose 5 dollars if neither coin is a head. Calculate the expected value of this game and determine whether it is favorable for the player.

Exercise 6

Knowing that p(x \le 2) = 0.7 and p(x \ge  2) = 0.75. Calculate:

1. The expected value.

2.The variance.

3.The standard deviation.

 

 

Solution of exercise 1

A pair of die is thrown. The random variable, X, is defined as the sum of the obtained scores. Determine the probability distribution, the expected value and variance.

 x { p }_{ i } x\quad .\quad { p }_{ i } { x }^{ 2 }\quad .\quad { p }_{ i }
2 \frac { 1 }{ 36 } \frac { 2 }{ 36 } \frac { 4 }{ 36 }
3 \frac { 2 }{ 36 } \frac { 6 }{ 36 } \frac { 18 }{ 36 }
4 \frac { 3 }{ 36 } \frac { 12 }{ 36 } \frac { 48 }{ 36 }
5 \frac { 4 }{ 36 } \frac { 20 }{ 36 } \frac { 100 }{ 36 }
6 \frac { 5 }{ 36 } \frac { 30 }{ 36 } \frac { 180 }{ 36 }
7 \frac { 6 }{ 36 } \frac { 42 }{ 36 } \frac { 294 }{ 36 }
8 \frac { 5 }{ 36 } \frac { 40 }{ 36 } \frac { 320 }{ 36 }
9 \frac { 4 }{ 36 } \frac { 36 }{ 36 } \frac { 324 }{ 36 }
10 \frac { 3 }{ 36 } \frac { 30 }{ 36 } \frac { 300 }{ 36 }
11 \frac { 2 }{ 36 } \frac { 22 }{ 36 } \frac { 242 }{ 36 }
12 \frac { 1 }{ 36 } \frac { 12 }{ 36 } \frac { 144 }{ 36 }
7 54.83

 

\mu =7

\sigma =\sqrt { 54.83-{ 7 }^{ 2 } }

\sigma =\sqrt { 54.83-49 }

\sigma =\sqrt { 5.83 }

\sigma =2.415

Solution of exercise 2

A player throws a die. If a prime number is obtained, he gains to win an amount equal to the number rolled times 100 dollars, but if a prime number is not obtained, he loses an amount equal to the number rolled times 100 dollars. Calculate the probability distribution and the expected value of the described game.

x { p }_{ i } x\quad .\quad { p }_{ i }
+100 \frac { 1 }{ 6 } \frac { 100 }{ 6 }
+ 200  \frac { 1 }{ 6 } \frac { 200 }{ 6 }
+ 300  \frac { 1 }{ 6 } \frac { 300 }{ 6 }
- 400  \frac { 1 }{ 6 } \frac { -400 }{ 6 }
+ 500  \frac { 1 }{ 6 } \frac { 500 }{ 6 }
-600  \frac { 1 }{ 6 } \frac { -600 }{ 6 }
 \frac { 100 }{ 6 }

\mu =16.667

 

Solution of exercise 3

The first prize for a raffle is 5,000 dollars (with a probability of 0.001) and the second prize is 2,000 dollars (with a probability of 0.003). What is a fair price to pay for a single ticket in this raffle?

\mu =(5000)(0.001)+(2000)(0.003)

\mu =11 dollars

 

Solution of exercise 4

Let X be a discrete random variable whose probability distribution is as follows:

x { p }_{ i }
0 0,1
1 0,2
2 0,1
3 0,4
4 0,1
5 0,1

 

f\left( x \right) =\begin{cases} \begin{matrix} 0 & \qquad x<0 \\ 0.1 & \qquad 0\le x<1 \\ 0.3 & \qquad 1\le x<2 \end{matrix} \\ \begin{matrix} 0.4 & \qquad 2\le x<3 \\ 0.8 & \qquad 3\le x<4 \\ 0.9 & \qquad 4\le x<5 \end{matrix} \\ \begin{matrix} 1 & \qquad \qquad 5\le x \end{matrix} \end{cases}

 

1. Calculate the distribution function.

2. Calculate the following probabilities:

p (x < 4.5)

p (x < 4.5) = F (4.5) = 0.9

 

p (x \ge 3)

p (x \ge 3) = 1 - p(X < 3) = 1 - 0.4 = 0.6

 

p (3 \le x < 4.5)

p (3 \le x < 4.5) = p (X < 4.5) - p(X < 3) = 0.9 - 0.4 = 0.5

 

Solution of exercise 5

A player tosses two coins into the air. He wins 1 dollar for the number of heads he will get. However, he will lose 5 dollars if neither coin is a head. Calculate the expected value of this game and determine whether it is favorable for the player.

E = {(h,h);(h,t);(t,h);(t,t)}

Probablity of getting 1 head= p(+1) = 2/4

Probablity of getting 2 heads= p(+2) = 1/4

Probablity of getting two tails= p(−5) = 1/4

 

\mu =\sum { { x }_{ i }.{ p }_{ i } }

\mu =(1)(\frac { 2 }{ 4 } )+(2)(\frac { 1 }{ 4 } )-(5)(\frac { 1 }{ 4 } )

\mu =-\frac { 1 }{ 4 }

Hence, it is unfavorable.

 

Solution of exercise 6

Knowing that p(x \le 2) = 0.7 and p(x \ge  2) = 0.75. Calculate:

1. The expected value.

2.The variance.

3.The standard deviation.

 

f\left( x \right) =\begin{cases} 0\qquad x<0 \\ 0.1\qquad 0\le x<1 \\ 0.1+a\qquad 1\le x<2 \\ 0.1+a+b\qquad 2\le x<3 \\ 0.1+a+b+c\qquad 3\le x<4 \\ 0.1+a+b+c+0.2\qquad 4\le x \end{cases}

p (x \le 2)

p (x \le 2) = 0.1+a+b = 0.7

 

p (x \ge 2)

p (x \ge 2) =  1 - (0.1-a) = 0.75

 

After solving the above equations simultaneously, the value of a will be "0 " and the value of b will be "0.45".

0.1+a+b+c+0.2=1

c=0.1

 x { p }_{ i } x\quad .\quad { p }_{ i } { x }^{ 2 }\quad .\quad { p }_{ i }
0 0.1 0 0
1 0.15 0.15 0.15
2 0.45 0.9 1.8
3 0.1 0.3 0.9
4 0.2 0.8 3.2
2.15 6.05

\mu =2.15

 

{ \sigma }^{ 2 }=6.05-{ 2.15 }^{ 2 }=1.4275

\sigma =1.19

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.