The continuous random variable TTT represents the deviation, in micrometres, of a high-precision digital caliper reading. The distribution of TTT is a continuous uniform distribution over the interval [0,2.5][0, 2.5][0,2.5].
Determine P(T<1.0)P(T < 1.0)P(T<1.0).
State the value of E(T)E(T)E(T).
Calculate Var(T)Var(T)Var(T).
A random sample of 20 readings taken by this caliper is recorded.
Find the probability that fewer than 9 of these readings show a deviation of more than 1.0 μm\mu \text{m}μm.
For a different model of caliper, the deviation X μmX \, \mu \text{m}Xμm is represented by the cumulative distribution function F(x)F(x)F(x) defined by:
F(x)={0x<00.8x−0.16x20≤x≤2.51x>2.5 F(x) = \begin{cases} 0 & x < 0 \\ 0.8x - 0.16x^2 & 0 \le x \le 2.5 \\ 1 & x > 2.5 \end{cases} F(x)=⎩⎨⎧00.8x−0.16x21x<00≤x≤2.5x>2.5Using this model, find the value of P(X>1.0)P(X > 1.0)P(X>1.0).
A large batch of 200 readings is collected from this different caliper.
Using a suitable approximation, find the probability that at least 80 of these readings show a deviation of more than 1.0 μm\mu \text{m}μm.
