Let the probability density function of a random variable x be given as f(x) = ae-2|x| The value of ‘a’ is __________.
Concept:
The probability density function for any f(x) is defined as:
\(\rm \int_{-\infty}^{+ \infty} f(x) dx = 1\)
Modulus function is f(x) = |x| is defined as:
\(\rm f(x) = \left\{ \begin{matrix} \rm -x;& - \infty < \rm x < 0 \\\ \rm + x; & 0 < \rm x < \infty \end{matrix} \right\}\)
Calculation:
\(\rm \int_{-\infty}^{+ \infty} f(x) dx = 1\)
\(\rm \int_{-\infty}^{0} ae^{-2(-x)}dx + \int_0^\infty ae^{-2(+x)}dx = 1\)
\(\rm \frac{a}{2} (e^{2x})^0_{-\infty} + \frac{a}{2} (-e^{-2x})^\infty_0 = 1\)
\(\rm \frac{a}{2} \{e^0 - e^{-\infty}\} + \frac{a}{2} \{ -e^{-\infty} -(-e^0)\} = 1\)
\( {a \over 2}\)(1-0) + \( {a\over 2}\)(0+1) = 1
a = 1
Important Questions from Random Variables Basics
The length of time X, needed by an examinee of competition to complete a 1-hour exam, is a random variable with
PDF \(f(x)=\dfrac{6}{5}(x^2+x);0 \le x \le 1.\) , The value of F(0.5) is:If X follows a binomial distribution with n = 6 and \(p=\dfrac{1}{4}\) then the skewness of X is:
If the customers arrive in a shop in Poisson fashion with parameter λ, the fourth raw moment \(\mu_4^{'}\) for the inter-arrival time is:
A discrete random variable X has the probability functions as:
X
0
1
2
3
4
5
6
7
8
f(x)
K
2k
3k
5k
5k
4k
3k
2k
k
The value of E(X) is:What percentage of scores falls within three standard deviations from the mean for the normal variate?
