Consider the following function: Calculate the following probabilities1.

Calculate the probability of a single event

hich of the following valid probabilities represents an event that is impossible? −1 0 0.0001 1

(b)Find the probability that it is discolored.

A random variable X has a normal probability distribution N (0, 1). De-termine the value of c as a function of u and o such thatP (X ≥ c) = 9P (X < c).(a) Find the value of c.(b) Evaluate P ((X − µ)2 ≥ σ)

a) [2 marks] Plot the probability density function, clearly marking on your plot thevalues of f (x) when x = 0 and x = 2 and labelling both axes.(b) [2 marks] Explain and show why the value of c must equal 12 .(c) [2 marks] Find P (1 < X < 1.5).(d) [3 marks] Find P ({1 < X < 1.5}|{X > 0.5}).(e) [2 marks] Find P ({X > 0.5}|{1 < X < 1.5}).(f) [4 marks] Suppose that we have another random variable U that is uniformlydistributed with parameters 2 and 4. That is, the probability density function of Uis given byf (u) = 12, 2 ≤ u ≤ 4.Further, U and X are independent. In addition, someone has told us that therandom variable Y defined byY = X + Uhas expected value equal to 113 and variance equal to 59 . Based on this information,calculate the expected value and variance of X.