Prob. Distribution Fcn f(x) = 0, x<=0; a/sqrt[x], 0<x<=1; 0, x>1; find "a", calculate

P R Flare said:

Hi,
Could anyone tell me if my answers are right? Thank you in advance.
P.

---

1. Consider a random variable X with the following probability distribution function.

. . . . .\(\displaystyle f(x)\, =\, \begin{cases}0,&\mbox{if }\, x\, \leq\, 0\\ \dfrac{a}{\sqrt{x\,}},&\mbox{if }\, 0\, <\, x\, \leq\, 1\\ 0,&\mbox{if }\, x\, >\, 1\end{cases}\)

1.1 Determine the value of a.
1.2 Calculate the expected value of X.
1.3 Determine the distribution function of X.
1.4 Calculate the probability of obtaining more than 0.25 in three independent observations (that is, more than 0.25 in each one of them).

---

My answer to 1.1 will follow in the next post.

---

\(\displaystyle \displaystyle F(x)\, =\, \int_{-\infty}^{+\infty}\, f(x)\, dx\, =\, 1\)

Because \(\displaystyle f(x)\, =\, 0\) except when \(\displaystyle 0\, <\, x\, \leq\, 1,\) we can say:

. . .\(\displaystyle \displaystyle F(x)\, =\, \int_0^1\, f(x)\, dx\, =\, 1\)

. . . . .\(\displaystyle \displaystyle \Longleftrightarrow\, \int_0^1\, \dfrac{a}{\sqrt{x\,}}\, dx\, =\, 1\)

. . . . .\(\displaystyle \displaystyle \Longleftrightarrow\, a\, \cdot 2\sqrt{x\,}\bigg|_0^1\, =\, 1\)

. . . . .\(\displaystyle \displaystyle \Longleftrightarrow\, a\, \big[(2\, \cdot 1)\, -\, (2\, \cdot\, 0)\big]\, =\, 1\)

. . . . .\(\displaystyle \displaystyle \Longleftrightarrow\, a\, \cdot\, 2\, =\, 1\)

. . . . .\(\displaystyle \displaystyle \Longleftrightarrow\, a\, =\, \dfrac{1}{2}\)

Click to expand...