Chapters

## 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.

## 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:

1. Calculate the distribution function.

2. Calculate the following probabilities:

## 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 and . 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.

## 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.

## 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?

dollars

## Solution of exercise 4

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

**1. Calculate the distribution function.**

**2. Calculate the following probabilities:**

## 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.

Probablity of getting 1 head=

Probablity of getting 2 heads=

Probablity of getting two tails=

Hence, it is unfavorable.

## Solution of exercise 6

Knowing that and . Calculate:

1. The expected value.

2.The variance.

3.The standard deviation.

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

The platform that connects tutors and students

exercise 6 is misleading. Many-many different r.v.’s fulfil the conditions, e.g.

P(X=1)=0.25, P(X=2)=0.45, P(X=3)=0.3.

Here value 1 can be any num less than 2, and value 3 can be any number greater than 2.

Very helpful resource. Thank you. But for exercise 2, 1 is not a prime, therefore it should be -100 instead of +100 which makes the answer -16.667 instead of 16.667