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In a gambling game a man wins Rs.$10 $ if he gets all heads or all tails and loses Rs.$5$ if he gets $1$ or $2$ heads when $3$ coins are tossed once. Find his expectation of gain.

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Toolbox:
  • If S is a sample space with a probability measure and X is a real valued function defined over the elements of S, then X is called a random variable.
  • Types of Random variables :
  • (1) Discrete Random variable (2) Continuous Random variable
  • Discrete Random Variable :If a random variable takes only a finite or a countable number of values, it is called a discrete random variable.
  • Continuous Random Variable :A Random Variable X is said to be continuous if it can take all possible values between certain given limits. i.e., X is said to be continuous if its values cannot be put in 1 − 1 correspondence with N, the set of Natural numbers.
  • The probability mass function (a discrete probability function) $P(x)$ is a function that satisfies the following properties :
  • (1) $P(X=x)=P(x)=P_x$
  • (2) $P(x)\geq 0$ for all real $x$
  • (3) $\sum P_i=1$
  • Moments of a discrete random variable :
  • (i) About the origin : $\mu_r'=E(X^r)=\sum P_ix_i^{\Large r}$
  • First moment : $\mu_1'=E(X)=\sum P_ix_i$
  • Second moment : $\mu_2'=E(X^2)=\sum P_ix_i^2$
  • (ii) About the mean : $\mu_n=E(X-\bar{X})^n=\sum (x_i-\bar{x})^nP_i$
  • First moment : $\mu_1=0$
  • Second moment : $\mu_2=E(X-\bar{X})^2=E(X^2)-[E(X)]^2=\mu_2'-(\mu_1')^2$
  • $\mu_2=Var(X)$
Step 1:
Let $X$ be the random variable denoting the amount that a man can win when 3 coins are tossed once.
$X$ takes the values Rs.10,-Rs.5
Step 2:
$P(X=10)$=Probability of all heads or all tails
$\qquad\quad\;\;\;=2\times (\large\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2})$
$\qquad\quad\;\;\;=\large\frac{1}{4}$
$P(X=-5)=1-\large\frac{1}{4}=\frac{3}{4}$
Step 3:
The probability distribution of $x$ is given by
Step 4:
$E(X)$=expectation of gain =$\sum x_iP_i$
$\qquad=10\times \large\frac{1}{4}$$+(-5)\times \large\frac{3}{4}$
$\qquad=\large\frac{10-15}{4}$
$\qquad=\large\frac{-5}{4}$
$\qquad=-1.25$
He is expected to lose Rs.1.25
answered Sep 17, 2013 by sreemathi.v
 

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