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 square of a standard normal variate is a
Let X1 and X2 be two independent exponentially distributed random variables with means 0.5 and 0.25, respectively. Then Y = min (X1, X2) is
An automobile plant contracted to buy shock absorbers from two suppliers X and Y. X supplies 60% and Y supplies 40% of the shod absorbers. All shock absorbers are subjected to a quality test. The ones that pass the quality test are considered reliable Of X's shock absorbers, 96% are reliable. Of Y's shock absorbers, 72% are reliable. The probability that a randomly chosen shock absorber, which is found to be reliable is made by Y is
For the function f(x) = a + bx, 0 ≤ x ≤ 1, to be a valid probability density function, which one of the following statements is correct?
If f(𝑥) and g(𝑥) are two probability density functions,
\(\begin{array}{l} f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{x}{a} + 1}&{: - a \le x < 0}\\ { - \frac{x}{a} + 1}&{0 \le x \le a}\\ 0&{otherwise} \end{array}} \right.\\ g\left( x \right) = \left\{ {\begin{array}{*{20}{c}} { - \frac{x}{a}}&{:-a \le x \le 0}\\ {\frac{x}{a}}&{:0 \le x \le a}\\ 0&{:otherewise} \end{array}} \right. \end{array}\)
Which one of the following statements is true?
