Consider a discrete random variable X whose probabilities are given below. The standard deviation of the random variable is ________ (round off to one decimal place).$x_1$ 1 2 3 4 $P(X = x_i)$ 0.3 0.1 0.3 0.3
Calculating Standard Deviation for a Discrete Random Variable
To find the standard deviation of the discrete random variable X, we first need to calculate its expected value (mean) and variance.
Probability Distribution Table
| $x_i$ | 1 | 2 | 3 | 4 |
| $P(X = x_i)$ | 0.3 | 0.1 | 0.3 | 0.3 |
Step 1: Calculate Expected Value (Mean, $E[X]$)
The expected value is calculated as the sum of each value multiplied by its probability.
$ E[X] = \sum_{i=1}^{n} x_i P(X = x_i) $
Using the given data:
$ E[X] = (1 \times 0.3) + (2 \times 0.1) + (3 \times 0.3) + (4 \times 0.3) $
$ E[X] = 0.3 + 0.2 + 0.9 + 1.2 $
$ E[X] = 2.6 $
Step 2: Calculate Expected Value of $X^2$ ($E[X^2]$)
This is needed to calculate the variance. It's the sum of each squared value multiplied by its probability.
$ E[X^2] = \sum_{i=1}^{n} x_i^2 P(X = x_i) $
Using the given data:
$ E[X^2] = (1^2 \times 0.3) + (2^2 \times 0.1) + (3^2 \times 0.3) + (4^2 \times 0.3) $
$ E[X^2] = (1 \times 0.3) + (4 \times 0.1) + (9 \times 0.3) + (16 \times 0.3) $
$ E[X^2] = 0.3 + 0.4 + 2.7 + 4.8 $
$ E[X^2] = 8.2 $
Step 3: Calculate Variance ($Var(X)$)
The variance is calculated using the formula:
$ Var(X) = E[X^2] - (E[X])^2 $
Substituting the calculated values:
$ Var(X) = 8.2 - (2.6)^2 $
$ Var(X) = 8.2 - 6.76 $
$ Var(X) = 1.44 $
Step 4: Calculate Standard Deviation ($SD(X)$)
The standard deviation is the square root of the variance.
$ SD(X) = \sqrt{Var(X)} $
$ SD(X) = \sqrt{1.44} $
$ SD(X) = 1.2 $
Rounding to one decimal place, the standard deviation is 1.2.
Important Questions from Random Variables
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?- Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained.
The expectation of $X$ is __________ (rounded off to two decimal places). - 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
If $X$ is a continuous random variable with the probability density function
$f(x) = \begin{cases} \frac{K}{4}, & 0 \le x \le 1 \\ 0, & \text{otherwise} \end{cases}$
then the value of $K$ is __________. (Answer in integer)
