Question Q-5: [7+5+8 Marks] Consider the random variable \( \mathrm{X} \) that has the cumulative distribution function \( \mathrm{F}(\mathrm{x})= \) \[ \begin{array}{ll} 0 & \mathrm{x}<0 \\ \frac{x^{2}}{4} & 0 \leq \mathrm{x}<2 \\ 1 & \mathrm{x} \geq 2 \end{array} \] a) Find the probability density function. b) Find \( \mathrm{P}(\mathrm{X}<1) \) c) Find the mean of Q-5: [7+5+8 Marks] Consider the random variable X that has the cumulative distribution function F(x)= 04x21x<00≤x<2x≥2 a) Find the probability density function. b) Find P(X<1) c) Find the mean of the random variable X.

2ZFOWU The Asker · Probability and Statistics

Transcribed Image Text: Q-5: [7+5+8 Marks] Consider the random variable X that has the cumulative distribution function F(x)= 04x21x<00≤x<2x≥2 a) Find the probability density function. b) Find P(X<1) c) Find the mean of the random variable X.

More

Community Answer

Honor Code

Solved 1 Answer

F(x)=\left\{\begin{array}{ccc}x^{2} / 4 & ; & 0 \leq x<2 \\1 & ; & x \geq 2\end{array}\right.a)\begin{array}{rlrl}f(x) & =f^{\prime}(x) & \\& =\left\{\begin{array}{ccc}0 & i & x<0 \\x / 2 & ; & 0 \leq x<2 \\0 & ; & x \geq 2\end{array}\right.\end{array}\begin{aligned}\text { s. } f(x)= & 0 \quad x \quad x<0 \\\frac{x}{2} & ; \quad 0 \leq x<2\end{aligned}b)\begin{aligned}P(x<1)= & \int_{0}^{1} \frac{x}{2} d x \\\rightarrow & {\left[\frac{x^{2}}{4}\right]_{0}^{1}=\frac{1}{4}=0.25 }\end{aligned}c)\begin{aligned}\varepsilon(x) & =\int_{0}^{2} \frac{x^{2}}{2} d x \\& \rightarrow\left[\frac{x^{3}}{6}\right]_{0}^{2}=\frac{8}{6}=1.3333\end{aligned}CS Scanned with CamScanner ...