Theory of Expectation :: Problems on Tossing Coins : Probability Distribution

Problem Back to Problems Page
 
A person tosses a coin and is to receive Rs. 4 for a head and is to pay Rs. 2 for a tail. Find expectation and variance of his gains.

Net Answers :
[Expectation: 1; Variance: 9]

Solution  
 

Let "x" indicate the amount gained by the person

[Since you are required to find the expectation of the players gain, the variable would represent the players gain]

The amount that the player can gain would be

  • + Rs. 4 if the coin shows up a head
  • − Rs. 2 if the coin shows up a tail

⇒ The values carried by the variable ("x") would be either + 4 or − 2
⇒ "X" is a discrete random variable with range = {− 2, + 4}

"X" represents the random variable and P(X = x) represents the probability that the value within the range of the random variable is a specified value of "x"

Probabilty that the coin shows up a

  • Head =
    1
    2
  • Tail =
    1
    2
    Since, in the experiment of tossing a coin there are two elementary events which are equally likely, mutually exclusive and exhaustive, the probability of occurance of each elementary event is ½ ⇒ P(HEAD) = P(TAIL) = ½

    Probability for the persons gain to be

  • + Rs. 4 ⇒ P(X = + 4) =
    1
    2
  • − Rs. 2 ⇒ P(X = − 2) =
    1
    2

    The probabilty distribution of "x" would be
    x − Rs. 2 + 4
    P(X = x)
    1
    2
    1
    2

    Calculations for Mean and Standard Deviations

    x P (X = x) px
    [x × P (X = x)]
    x2 px2
    [x2 × P (X = x)]
    − 2
    1
    2
    − 2
    2
    4
    4
    2
    + 4
    1
    2
    + 4
    2
    16
    16
    2
    Total 1
    + 2
    2
    20
    2
    = + 1 = + 10

    The persons expected gain


    ⇒ Expectation of "x"
    ⇒ E (x) = Σ px
    = 1
    Variance of the Persons gain
    ⇒ var (x) = E (x2) − (E(x))2
    ⇒ var (x) = Σ px2 − (Σ px)2
    = 10 − (1)2
    = 10 − 1
    = 9
    Standard Deviation of the persons gain
    ⇒ SD (x) = + Var (x)
    = + 9
    = + 3

    Credit : Vijayalakshmi Desu

    © Krishbhavara ♣