Probability Question

[nacho]

The time, X seconds, for a fast-acting anaesthetic to put a patient to sleep is a continuous random variable with a probability density function f(x) = k(8x-x^2), 1<x<8, and zero elsewhere.
so far,
i've shown k = 3/245.

a) find the average time in seconds which the aneasthetic takes to put a patient to sleep.
b) find the median of the probability density function correct to 4 d.p
c) find the most common time, in seconds, that the anaesthetic takes to put a patient to sleep
d. calculate SD(X)
e. find the probability, correct to 3 d.p that a patient will be put to sleep within on SD of the mean (2 marks)

okay,
for a) i got 4.17s
b) 4.1144 s
c) i dont know how to do
d) how does one work out E(X^2) ?
e) i need part d

[tony3272]

The time, X seconds, for a fast-acting anaesthetic to put a patient to sleep is a continuous random variable with a probability density function f(x) = k(8x-x^2), 1<x<8, and zero elsewhere.
so far,
i've shown k = 3/245.

a) find the average time in seconds which the aneasthetic takes to put a patient to sleep.
b) find the median of the probability density function correct to 4 d.p
c) find the most common time, in seconds, that the anaesthetic takes to put a patient to sleep
d. calculate SD(X)
e. find the probability, correct to 3 d.p that a patient will be put to sleep within on SD of the mean (2 marks)

okay,
for a) i got 4.17s
b) 4.1144 s
c) i dont know how to do
d) how does one work out E(X^2) ?
e) i need part d

c. The most common time one event occurs just refers to the mode. So just solve for the maximum, in this case at x=4.

d.