Probstat/week8 practice 1

This is part of probstat

This exercise considers continuous random variables.

Probability

1. Let random variable $X$ is a continuous random variable which is a real number chosen uniformly from the range (10,50). Let $f(x)$ denote its probability distribution function.

• 1.1 Describe $f(x)$.
• 1.2 What is $P\{X>20\}$.
• 1.3 What is $P\{X>20|X<40\}$. (Note that this is a conditional probability.)

2. Let random variable $X$ is a continuous random variable which is a real number chosen uniformly from the range (a,b). Let $f(x)$ denote its probability distribution function.

• 2.1 Describe $f(x)$.
• 2.2 Find the value of $c$ such that $P\{X>c\}=1/2$.

Definition: A cumulative distribution function (or cdf in short) for a continuous random variable $X$whose probability distribution function is $f(x)$ is defined to be

$F(a)=P\{X\leq a\}=P\{X<a\}=\int _{-\infty }^{a}f(x)dx.$

3. Let random variable $X$ is a continuous random variable whose probability distribution function $f(x)$ is given by

$f(x)=\left\{{\begin{array}{ll}cx\ \ \ \ &{\mbox{if}}\ 0\leq x\leq 20,\\0&{\mbox{otherwise.}}\end{array}}\right.$

• 3.1 Find the value of c.
• 3.2 Plot pdf of X.
• 3.3 Let $F(x)$ be $X$'s cumulative distribution function. Find $F(x)$.
• 3.4 Plot $F(x)$

4. (FCP. Example 1a.) Suppose that X is a continuous random variable whose pdf is given by

$f(x)=\left\{{\begin{array}{ll}C(9x-3x^{2})\ \ \ &0<x<3\\0&{\mbox{otherwise.}}\end{array}}\right.$

• 4.1 Find $C$.
• 4.2 Find $P\{X>1\}$.

5. (FCP. Example 1b.) The time (in hours) that a computer functions before breaking down is a continuous random variable $X$ with probability density function given by

$f(x)=\left\{{\begin{array}{ll}\lambda e^{-x/100}\ \ \ &x\geq 0\\0&x<0\end{array}}\right.$

• 5.1 Find the value of $\lambda$
• 5.2 What is the probability that a computer will function between 50 and 100 hours before breaking down?
• 5.3 What is the probability that it will function less than 100 hours?

Hint: Note that

$\int e^{-x}dx=-e^{-x}+C$

6. A random variable $X$ has pdf defined by

$f(x)=\left\{{\begin{array}{ll}C/x^{2}\ \ \ &x>20\\0&{\mbox{otherwise.}}\end{array}}\right.$

• 6.1 Find $C$.
• 6.2 Find its cdf $F(x)$ and plot it.
• 6.3 What is $P\{X>30\}$?

Expectation

Definition: If $X$ is a continuous random variable whose probability distribution function (pdf) is $f(x)$, we define the expected value of $X$ to be

$\mathrm {E} [X]=\int _{-\infty }^{\infty }xf(x)dx$

1. Find $\mathrm {E} [X]$ for continuous random variable $X$ defined in question 1 in the previous section.

2. Find $\mathrm {E} [X]$ for continuous random variable $X$ defined in question 2 in the previous section.

3. Find $\mathrm {E} [X]$ for continuous random variable $X$ defined in question 3 in the previous section.

4. Find $\mathrm {E} [X]$ for continuous random variable $X$ defined in question 4 in the previous section.

5. Find $\mathrm {E} [X]$ for continuous random variable $X$ defined in question 5 in the previous section.

Hint: Note that

$\int xe^{-x}dx=-e^{-x}(x+1)+C$

6. Find $\mathrm {E} [X]$ for continuous random variable $X$ defined in question 6 in the previous section.

7. Find $\mathrm {E} [X^{2}]$ and Var($X$) for continuous random variable $X$ defined in question 1 in the previous section.

8. Find $\mathrm {E} [X^{2}]$ and Var($X$) for continuous random variable $X$ defined in question 2 in the previous section.

9. Find $\mathrm {E} [X^{2}]$ and Var($X$) for continuous random variable $X$ defined in question 3 in the previous section.