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
Suppose X is a continuous random variable with probability density function
\(f(x)=\frac{1}{\pi} \frac{1}{1+(x+1)^2}\), -∞ < x < ∞.
Define
\(Y=\left\{\begin{array}{cc} \frac{X}{|X|}, & \text { if } X \neq 0 \\ 0, & \text { if } X=0 \end{array}\right.\)
Then which of the following statements are true?
Let X1, X2, ..., Xn be a random sample from an absolutely continuous distribution with the probability density function
\(f(x \mid \theta)=\left\{\begin{array}{cl} e^{\theta-x}, & \text { if } x \geq \theta \\ 0, & \text { if } x<\theta \end{array},\right.\)
where θ ∈ ℝ is unknown. Define \(\bar{X}=\frac{1}{n} \sum_{i=1}^n X_i\) and X(1) = min{X1, ..., Xn}. Then
which of the following statements are true?
Suppose that X is a continuous random variable with probability density function given by:
f(x) = \(\left\{ {\begin{array}{c} {\frac{x}{8},}&{x \in \left[ {0,2} \right)}\\ {\frac{1}{4},}&{x \in \left[ {2,4} \right)}\\ { - \frac{x}{8} + \frac{3}{4},}&{x \in \left[ {4,6} \right)} \end{array}}\right.\)
Find the mean of X.
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 _________
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
