Question

Let Y ~ Unif(1,5).

1.If you generate 5 random numbers based on Y, what is the probability you'll get more(numbers greater than 4) than (numbers less than or equal to 4)?

2.If you take lots of random values for Y and plug them into the polynomial h(x) = (2x−1)(x+ 3),what value would you get out of the polynomial on average?

Answer 1

We are given the distribution here as:

1) The probability that number greater than 4 comes is computed here as:

The number of numbers greater than 4 obtained in the 5 numbers generated could be modelled here as:

The probability that you'll get more(numbers greater than 4) than (numbers less than or equal to 4) is computed here as:

P(X = 3) + P(X = 4) + P(X = 5)

Therefore 0.1035 is the required probability here.

b) The expected value of Y and the second moment of Y are first computed here as:

E(Y) = (a + b)/2 = (1 + 5)/2 = 3

Var(Y) = (b - a)2 / 12 = (5 - 1)2 /12 = 4/3

Therefore E(Y2) = Var(Y) + [E(Y)]2 = (4/3) + 32 = 10.3333

Now the expected value of h(x) is computed here as:

E[ h(x)] = E [ (2x - 1)(x + 3) ] = E [ 2x2 + 5x - 3] = 2E(X2) + 5E(X) - 3

= 2*10.3333 + 5*3 - 3 = 32.6667

Therefore 32.6667 is the required expected value of the polynomial here.