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# A die is tossed twice. A success is getting as odd number on a toss. Find the mean and the variance of the probability distribution of the number of successes.

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 number of times an odd number (the number of successes) when a die is tossed twice.
$x$ takes the values 0,1,2
Step 2:
P(X=0)=probability of no odd number(both dice showing even)
$\Rightarrow \large\frac{3}{6}\times \frac{3}{6}=\frac{1}{4}$
P(X=1)=probability of one odd number
$\Rightarrow 2C_1\large\frac{3}{6}\times\frac{3}{6}=\frac{1}{2}$
P(X=2)=probability of both dice showing odd numbers
$\Rightarrow \large\frac{3}{6}\times\frac{3}{6}=\frac{1}{4}$
Step 3:
The probability distribution of $X$ is given by
Step 4:
$E(X)=\sum x_iP_i=0\times \large\frac{1}{4}$$+1\times \large\frac{1}{2}$$+2\times \large\frac{1}{4}=1$
Step 5:
$var(X)=E(X^2)-[E(X)]^2$
$E(X^2)=\sum x_i^2P_i=0\times \large\frac{1}{4}$$+1\large\frac{1}{2}$$+4\times \large\frac{1}{4}$
$\Rightarrow \large\frac{3}{2}$