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
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 :
The density function of a random variable is given by \(p(x)=Ke^{-\frac{x^{2}}{2}}\) for \(-\infty \lt x \lt \infty\). The value of K should be :
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
