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 $X$, $N$, $Y$ and $Z$ be random variables. The variables $X$ and $N$ are independent of each other. $X$ is uniformly distributed between -1 and 1; $N$ follows Normal distribution with zero mean and unity variance.
$Y$ and $Z$ are defined as, $Y = X + N$ and $Z = X^2 + N$.
Which of the following pairs represents the values of correlation between $X$ and $Y$ and that between $X$ and $Z$? Consider the following probability density function for a random variable x.
(a) \(f_{1}(x)=1\ ;\ -1\le x\le 1\)
(b) \(f_{2}(x)=x\ ;\ 0\le x\le 1\)
(c) \(f_{3}(x)=(1-x)\ ;\ -1\le x\le 1\)
Arrange the above functions in terms of the increasing value of mean of random variable x.
Considering all the symbols with their usual meanings, match the following :
List - I List - II (a) \(f_{X}(x)\) (i) \(\displaystyle\int_{-\infty}^{\infty}f_{X,Y}(x,y)\,dx\) (b) \(\displaystyle\int_{-\infty}^{\infty}f_{X}(x)\,dx\) (ii) \(\displaystyle\int_{-\infty}^{a}f_{X}(x)\,dx\) (c) \(F_{X}(a)\) (iii) \(\displaystyle\int_{-\infty}^{\infty}f_{X,Y}(x,y)\,dy\) (d) \(f_{Y}(y)\) (iv) 1 Codes :
