Each time a button is pressed, a machine delivers a plate of plastic material having a thickness which is distributed (each time, and independently of the previous ones) as an exponential random variable of parameter = 2. Pressing the key once, one obtains a plate of thickness X . If X > 1/2 he can sell the plate making a profit of 1 euro, otherwise, he can press the button a second time to get a new plate of thickness X. If X > 1/2 he can sell the new plate making a profit of only 0.50 euros. If even the second plate is too thin, he cannot continue, the residual plastic material is thrown away and nothing is gained.

Which of the following statements are True, and which are false, Just tell me true/false,

1) (Let p the probability of a gain of 1 euro. Then p=e^(-)

2) The probability of a gain of 0,50 euro is 2p(1 - p)

3) The expected gain is equal 1/2p(3- p)

4) The probability that the second plate thickness is greater than 1.5 knowing that 0.50 euros have been gained is e^(-2)

Now Assuming that the button has been pressed twice, now take the thicknesses X, and X of the two plates dispensed by the machine and add them to obtain an overall thickness X = X + X, of average and variance ^2

5) P(X 1) is equal to 1 - 3e^(-2)

6) A comparison is made between the overall thickness X and the thickness Y of a plate produced in another department, which is distributed as Y ~N( ,^2). Then P(Y) > P(X )