91影视

Compute the hazard rate function of a Weibull random variable and show it is increasing when $饾浗鈮�1$and decreasing when $饾浗鈮�1$

One thousand independent rolls of a fair die will be made. Compute an approximation to the probability that the number $6$will appear between $150$and $200$times inclusively. If the number $6$appears exactly $200$times, find the probability that the number 5 will appear less than $150$times.

The lifetimes of interactive computer chips produced

by a certain semiconductor manufacturer are normally distributed with parameters$渭=1.4脳{10}^6$hours and $蟽=3脳{10}^5$hours. What is the approximate probability that abatch of $100$chips will contain at least $20$whose lifetimes are less than $1.8脳{10}^6$?

Show that a plot of $\log \left(\log (1-\mathrm{F}(\mathrm{x}){)}^{-1}\right)$against $\log \left(\mathrm{X}\right)$will be a straight line with slope $\mathrm{尾}$when $\mathrm{F}(-)$is a Weibull distribution function. Show also that approximately $63.2$percent of all observations from such a distribution will be less than $\mathrm{伪}$. Assume that $\mathrm{v}=0$.

Let $Y=\frac{X-饾湀}{饾浖}$.

Show that if X is a Weibull random variable with parameters 谓, 伪, and 尾, then Y is an exponential random variable with parameter 位 = 1 and vice versa.

If $\mathrm{x}$is a beta random variable with parameters $\mathrm{a}$and $\mathrm{b}$, show that

$\mathrm{E}\left[\mathrm{X}\right]=\frac{\mathrm{a}}{\mathrm{a}+\mathrm{b}}$

$Var\left(\mathrm{X}\right)=\frac{\mathrm{ab}}{(\mathrm{a}+\mathrm{b}{)}^2(\mathrm{a}+\mathrm{b}+1)}$

If $\mathrm{x}$is uniformly distributed over $(\mathrm{a},\mathrm{b})$, what random variable, having a linear relation with $\mathrm{x}$, is uniformly distributed over $(0,1)$$?$

In $10,000$independent tosses of a coin, the coin landed on heads $5800$times. Is it reasonable to assume that the coin is not fair? Explain.

Twelve percent of the population is left handed. Approximate the probability that there are at least $20$ lefthanders in a school of $200$ students. State your assumptions.

Consider the beta distribution with parameters $(\mathrm{a},\mathrm{b})$. Show that

(a) when $\mathrm{a}>1$and $\mathrm{b}>1$, the density is unimodal (that is, it has a unique mode) with mode equal to $(\mathrm{a}-1)/(\mathrm{a}+\mathrm{b}-2)$

(b) when $\mathrm{a}鈮�1$, $\mathrm{b}鈮�1$, and $\mathrm{a}+\mathrm{b}<2$, the density is either unimodal with mode at $0$or $1$or U-shaped with modes at both$0$and$1$;

(c) when $\mathrm{a}=1=\mathrm{b}$, all points in $[0,1]$are modes.