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mathematics UK Year 12 Mathematics (AS)
 
Solved problems for the chapter:
 

Expectation and variance of a discrete random variable

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https://en.openprof.com/wb/chapter:expectation_and_variance_of_a_discrete_random_variable/5451/


Example 1.

Two fair 6-sided dice with faces labelled 1, 2, 3, 4, 5, 6 are thrown. The two scores are noted. The random variable is defined as follows.


  • If the two scores are equal,


  • If the scores are not equal, is the larger score minus the smaller score


Find and .

    
Example 2.

Jacob throws three coins at the same time. The first coin is biased so that the probability of obtaining a head when it is thrown is . The second coin is biased so that the probability of obtaining a head when it is thrown is . The third coin is biased so that the probability of obtaining a head when it is thrown is . The random variable X is the number of heads obtained. Given that , find .

    
3.

The random variable takes the value with probability , where is a constant and takes the values only. Find .

    
4.

Kayla has a bag containing 3 red marbles, 1 blue marble and 2 green marbles. She selects one marble from the bag at random and does not replace it in the bag. She repeats this process until she obtains a green marble. The random variable X is the number of marbles that she needs to select until she obtains a green marble. Find .

    
5.

A fair red spinner has 4 sides, numbered 1, 2, 3, 4. A fair blue spinner has 4 sides, numbered 0, 1, 2, 3. When a spinner is spun, the score is the number on the side on which it lands. The two spinners are spun at the same time. The random variable denotes the higher of the two scores obtained. If the two scores are equal, then the value of is 0. Find .

    
6.

Kai has a spinner with four sides, labelled 1, 2, 3, 4. When the spinner is spun, the score is the number on the side on which the spinner lands. The random variable denotes this score. The probability distribution table for is given below.




Find the numerical value of .

    
7.

The random variable takes the values . It is given that , where is a positive constant. Find the value of .

    
8.

A fair coin and an ordinary fair six-sided dice are thrown at the same time. The random variable is defined as follows.


  • If the coin shows a tail, is twice the score on the dice.

  • If the coin shows a head, is the score on the dice if the score is even and is 0 otherwise.


Find .

    

9.

A red fair six-sided dice has faces labelled 1, 1, 1, 2, 2, 2. A blue fair six-sided dice has faces labelled 1, 1, 2, 2, 3, 3. Both dice are thrown. The random variable X is the product of the scores on the two dice. Find .

    
10.

Jasmine has one coin, two coins and two coins. She selects two of these coins at random. The random variable is the total value, in dollars, of these two coins. Find the value of .