The expectation of $X$ is __________ (rounded off to two decimal places).
Calculating Expectation of Sum of Two Dice
Let $X_1$ be the random variable representing the outcome of the first fair die, and $X_2$ be the random variable representing the outcome of the second fair die. The possible outcomes for each die are $\{1, 2, 3, 4, 5, 6\}$.
The expectation (or mean) of a single fair die roll is calculated as:
$ E(X_1) = E(X_2) = \sum_{i=1}^{6} i \cdot P(X=i) $
Since the die is fair, the probability of rolling any face is $\frac{1}{6}$.
$ E(X_1) = \left( 1 \times \frac{1}{6} \right) + \left( 2 \times \frac{1}{6} \right) + \left( 3 \times \frac{1}{6} \right) + \left( 4 \times \frac{1}{6} \right) + \left( 5 \times \frac{1}{6} \right) + \left( 6 \times \frac{1}{6} \right) $
$ E(X_1) = \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5 $
Similarly, $E(X_2) = 3.5$.
Applying Linearity of Expectation
Let $X$ be the random variable denoting the sum of the outcomes, so $X = X_1 + X_2$.
Using the linearity of expectation, the expectation of the sum is the sum of the expectations:
$ E(X) = E(X_1 + X_2) = E(X_1) + E(X_2) $
$ E(X) = 3.5 + 3.5 = 7 $
Final Result
The expectation of $X$ is exactly 7. Rounded to two decimal places, the value is 7.00.
Important Questions from Random Variables
If the odds in favour of any random event A are 5 ∶ 6, then the odds against the event are:
If random variable X follows binomial distribution with parameter n and p with mean 15 and variance 10, then the value of mode is
Let $X$ and $Y$ be continuous random variables with probability density functions $P_X(x)$ and $P_Y(y)$, respectively. Further, let $Y = X^2$ and $P_X(x) = \begin{cases} 1, & x\in (0,1] \\ 0, & \text{otherwise} \end{cases}$
Which one of the following options is correct?- Let $X = aZ + b$, where $Z$ is a standard normal random variable, and $a, b$ are two unknown constants. It is given that
$E[X] = 1$, $E[(X – E[X])Z] = –2$, $E[(X – E[X])^2] = 4$,
where $E[X]$ denotes the expectation of random variable $X$. The values of $a, b$ are: Let $Y = Z^2$, $Z = \frac{X - \mu}{\sigma}$, where $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$. The variance of $Y$ is
