# Bsp Mid-Term

**Topics:**Random variable, Probability theory, Normal distribution

**Pages:**3 (323 words)

**Published:**February 7, 2013

STOCHASTIC

Tri 2’ 2012/2013

Mid-Term Exam

Question 1

a) A continuous random variable X has the probability density function given by 3

2

; 0 x2

(2 x)

f ( x) 8

0

; otherwise

(i)

Calculate the mean of X and variance of X.

(ii)

Calculate

.

(iii)

Find

.

b) Given X ~ Exp ( 2) and the moment generating function (MGF) of X is given

M X (t )

2

2t

.

Find the mean and variance of X.

c) Given

for x = 1, 2, 3, 4. Find the moment generating function of X.

Question 2

a) According to a survey, 45% of all students at a large university suffer from math anxiety. Four students are randomly selected from this university. Let X denotes the number of students in this sample who suffer from math anxiety.

(i)

Construct the probability distribution of X. [Leave your answer in 3 decimal places] (ii)

Find the probability of at least two students suffer from math anxiety. (iii)

Find the probability of 3 or 4 students suffer from math anxiety.

b) A research scientist reports that mice will live an average of 40 months when their diets are sharply restricted and then enriched with vitamins and proteins. Assuming that the lifetime of such mice is normally distributed with a variance of 3.308 year. (i)

Find the probability that a given mouse will live more than thirty two months. (ii)

Find the probability that a given mouse will live between forty two months and fifty months.

(iii)

Find the lifetime exceeded 85% of these mice.

c) A group of students consists of 60% men and 40% women. Among the men, 30% are foreign students, and among the women, 20% are foreign students.

(i)

What is the probability that a selected foreign student is a woman? (ii)

What is the probability that a selected student is a foreign man?

BSP2014

STOCHASTIC

Tri 2’ 2012/2013

End of Questions

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