Question

I what Is the probability that the cashier will claim it is fake? ii. what is the probability that the bill is in fact fake, given that the cashier identified it as such? (b) Let X be a continuous random variable with probability density function given by 15 otherwise 3(5-3r2) i. Show that the cumulative distribution function is F(x) - for 0 < <1. ii. Calculate P(X s 0.5). I ii. What is E[X]? 34 marks] 2. (a) According to researchers, a particular lake presents, on average, 3 species of arch

Answer 1

solution:

1.

$F(x) = \int_{0}^{x}\frac{15}{2} (t^2-t^4) dt$

$F(x) = \frac{15}{2}(\frac{t^3}{3}-\frac{t^5}{5})_0^{x}$

$F(x) = \frac{15}{2}(\frac{x^3}{3}-\frac{x^5}{5})$

$F(x) = \frac{x^3}{2}(5 - 3.x^2)$    for 0<x<1

  

2.

$P(X\leq 0.5)$

$F(0.5) = \frac{0.5^3}{2}(5 - 3*0.5^2)$

F (0.5) = 4.953

3.

$E(X) = \int_{0}^{1}x.f(x)dx$

$E(X) = \frac{15}{2}(\frac{x^4}{4}-\frac{x^5}{5})_0^1$

$E(X) = \frac{15}{2}(\frac{1}{4}-\frac{1}{5})$

$E(X) = \frac{3}{8}$