The processing time TTT (in microseconds) of a cryptographic hashing algorithm is modeled by a continuous random variable with probability density function:
f(t)={t200≤t≤k10−t30k<t≤100otherwise f(t) = \begin{cases} \frac{t}{20} & 0 \le t \le k \\ \frac{10-t}{30} & k < t \le 10 \\ 0 & \text{otherwise} \end{cases} f(t)=⎩⎨⎧20t3010−t00≤t≤kk<t≤10otherwiseBy considering the total area under the probability density function, show that k k\,k satisfies the equation k2−8k+16=0k^2 - 8k + 16 = 0k2−8k+16=0 and hence find the value of kkk.
Determine the mode of TTT.
Find P(T≤k2∣T≤k)\displaystyle P\left( T \le \frac{k}{2} \mid T \le k \right)P(T≤2k∣T≤k).
