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Probability is a branch of mathematics that studies the probability of a random event occurring. When a coin is tossed into the air, for example, the possible outcomes are Head and Tail.

(a) It is given that X is a random variable denoting the completion time for a certain task. Its CDF F(x) is given as:

Proposition: If X is a continuous rv with pdf f(x) and cdf F(x), then at every x at which the derivative\({F^\prime }(x)\)exists,

\(f(x) = {F^\prime }(x)\)

For intervals\(x < 0\)and\(x > \frac{7}{3}\):

\(f(x) = {F^\prime }(X) = 0\)

For interval\(0 \le x < 1\):

\(f(x) = \frac{d}{{dx}}\left( {\frac{{{x^3}}}{3}} \right) = {x^2}\)

For interval\(1 \le x \le \frac{7}{3}\)

\(\begin{array}{l}f(x) = \frac{d}{{dx}}\left( {1 - \frac{1}{2}\left( {\frac{7}{3} - x} \right)\left( {\frac{7}{4} - \frac{3}{4}x} \right)} \right)\\f(x) = \frac{1}{2}\left( {\frac{7}{4} - \frac{3}{4}x} \right) + \frac{1}{2}\left( {\frac{7}{3} - x} \right)\left( {\frac{3}{4}} \right)\\f(x) = \frac{7}{4} - \frac{3}{4}x\end{array}\)

Hence pdf f(x) can finally be given as:

\(f(x) = \left\{ {\begin{array}{*{20}{l}}{{x^2}}&{0 \le x < 1}\\{\frac{7}{4} - \left( {\frac{3}{4}} \right)x}&{1 \le x \le \frac{7}{3}}\\0&{{\rm{ otherwise }}}\end{array}} \right.\)

(b) compute\(P(0.5 < X < 2)\)

\(\begin{aligned}P(0.5 < X < 2) &= F(2) - F(0.5) \hfill \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &= \left[ {1 - \frac{1}{2}\left( {\frac{7}{3} - 2} \right)\left( {\frac{7}{4} - \frac{3}{4} \cdot 2} \right)} \right] - \left[ {\frac{{{{(0.5)}^3}}}{3}} \right] \hfill \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &= \frac{{23}}{{24}} -\frac{1}{{24}} \hfill \\ P(0.5 < X < 2) &= 0.9167 \hfill \\\end{aligned}\)

Proposition: Let\({\rm{X}}\)be a continuous\({\rm{rv}}\)with\({\rm{pdff}}({\rm{x}})\)and\(cdf\;{\rm{F}}({\rm{x}})\). Then for any two numbers a and\({\rm{b}}\)with\(a < b\),

\(P(a \le X \le b) = F(b) - F(a)\)

(c) The mean value of the given distribution can be given as:

\(\begin{aligned}E(X)&= \int_{ - \infty }^i\cdot f(x) \cdot dx \\ &= \int_0^1 x \cdot {x^2} \cdot dx + \int_1^{7/3} x \cdot \left( {\frac{7}{4} - \frac{3}{4}x} \right) \cdot dx \\&= \int_0^1 {{x^3}} \cdot dx + \int_1^{7/3} {\frac{{7x}}{4}} \cdot dx - \int_1^{7/3} {\frac{{3{x^2}}}{4}} \cdot dx \\&= \left[ {\frac{{{x^4}}}{4}} \right]_0^1 + \left[ {\frac{{7{x^2}}}{8}} \right]_1^{7/3} + \left[ {\frac{{{x^3}}}{4}} \right]_1^{7/3} = {\left[ {\frac{{{1^4}}}{4}} \right]_1} + \left[ {\frac{{343}}{{72}} - \frac{7}{8}} \right] + \left[ {\frac{{343}}{{108}} - \frac{1}{4}} \right] \\&= \frac{1}{4} + \frac{{35}}{9} - \frac{{316}}{{108}} \\E(X) &= 1.213 \\\end{aligned} \)

Definition: The expected or mean value of a continuous\(rv\;X\)with pdf f(x) is

\(\mu = E(X) = \int_{ - \infty }^\infty x \cdot f(x) \cdot dx\)