If the probability density function of a random variable x is given by \(f(x)=\left\{ {\begin{array}{*{20}{c}} {\frac{{k{x^2}}}{2}}&{ - 1 \le x \le 1}\\ 0&\rm {elsewhere,} \end{array}} \right.\) the value of k is ______.
Concepts:
For probability density function:
\(\int_{-\infty }^{\infty}f(x)dx = 1\)
where f(x) is probability density function
Calculation:
Given:
\(f(x)=\left\{ {\begin{array}{*{20}{c}} {\frac{{k{x^2}}}{2}}&{ - 1 \le x \le 1}\\ 0&\rm {elsehwhere,} \end{array}} \right.\)
\(\int_{-\infty }^{\infty}f(x)dx = 1\)
\(\int_{-\infty }^{-1}f(x)dx \ +\int_{-1}^{1}f(x)dx \ +\int_{1}^{\infty}f(x)dx = 1\)
\(\int_{-\infty }^{-1}0dx \ +\int_{-1}^{1}\frac{kx^2}{2}dx \ +\int_{1}^{\infty}0dx = 1\)
\(\frac{{k{x^3}}}{6}\left| {\begin{array}{*{20}{c}} 1\\ { - 1} \end{array}} \right.\) = 1
∴ k = 3
Important Questions from Continuous Distributions
Probability density function of a random variable X is given below
\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {0.25}&{if\;1 \le x \le 5}\\ 0&{otherwise} \end{array}} \right.\)
P (X ≤ 4) is
The variable x takes a value between 0 and 10 with uniform probability distribution. The variable y takes a value between 0 and 20 with uniform probability distribution. The probability of the sum of variables (x + y) being greater than 20 is _________
A nationalized bank has found that the daily balance available in its savings accounts follows a normal distribution with a mean of Rs. 500 and a standard deviation of Rs. 50. The percentage of savings account holders, who maintain an average daily balance more than Rs 500 is _______
The number of parameters in the univariate exponential and Gaussian distributions, respectively are
Find the value of λ such that the function f (x) is a valid probability density function. _______
\(f\left( x \right)\begin{array}{*{20}{c}} { = \lambda \left( {x - 1} \right)\left( {2 - x} \right)}&{for1 \le x \le 2}\\ { = 0}&{otherwise} \end{array}\)
