$($a$)$ An item is produced in large numbers. The machine is known to produce $10\%$

defectives. A quality control inspector is examining the items by taking them at

random. What is the probability that atleast $5$ items are to be examined to get $2$

defectives $.$

$($b$)$ Let $x$ be a random variable denoting number of trials required to get first success $,$

where trials are independent and Bernoullian each with same constant probability of

success $\text{'}p\text{'}.$ Show that for any two positive integers $s$ and $t$ :

$P[x>s+t|x>s]=P[x>t].$

$($c$)$ If $xU(0,1),$ find the distribution of $-2lo{g}_{e}x$ and identify it$.$ Hence obtain the

distribution of $-2lo{g}_{e}P,$ where $P=x_1{x}_{2}dots{x}_n,$ and $x_1,x_2,$dots,$x_n$ are iid $U(0,1)$

variates.