Standard Probability Distributions - Advanced Probability Theory for Biomedical Engineers - page 41

38. A 250 question multiple choice final exam is given. Each question has 5 possible answers
and only one correct answer. Determine the probability that a student guesses the correct
answers for 20-25 of 85 questions about which the student has no knowledge.
39. The average golf score for Professor Rensselaer is 78 with a standard deviation of 3.
Assuming a Gaussian distribution for random variable x describing her golf game,
determine: (a) P ( x 80), (d) the probability that

x is less than 75 if the score is measured to the nearest unit.
40. Suppose a system contains a component whose length is normally distributed with a
mean of 2.0 and a standard deviation of 0.2. If 5 of these components are removed from
different systems, what is the probability that at least 2 have a length greater than 2.1?
41. A large box contains 10,000 resistors with resistances that are Gaussian distributed. If
the average resistance is 1000 ohms with a standard deviation of 200 ohms, how many
resistors have resistances that are within 10% of the average?
42. The RV x has PDF

= 78), (b) P ( x ≤ 78), (c) P (70 < x ≤

a exp ) 2

1

f x ( α ) = −

2 (

α − η .

2 σ

(a) Find the constant a . (Hint: assume RV y is independent of x and has PDF f y ( β ) =

2
x

f x (

β ) and evaluate F x , y ( ∞ , ∞ ).) (b) Using direct integration, find E ( x ). (c) Find σ

using direct integration.
43. Assume x and y are jointly distributed Gaussian random variables with x ∼ G ( − 2 , 4),

y ∼ G (3 , 9), and ρ = 0. Find: (a) P (1 < y < 7 | x = 0), (b) P (1 < y < 7),

x , y

7).
44. Suppose x and y are jointly distributed Gaussian random variables with E ( y

(c) P ( − 1 < x < 1 , 1 < y <

| x ) =

2 . 8 + 0 . 32 x , E ( x | y ) =− 1 + 0 . 5 y , and σ = 3 . 67. Determine: (a) η

x , (b)

η

y , (c)

σ

x , y | x

(d) σ

y , (e)

ρ

y ,(f)

σ

y .

x , x ,

45. Assume x ∼ G (3 , 1), y ∼ G ( − 2 , 1), and that x and y are jointly Gaussian with ρ =

x , y

5. Draw a sketch of the joint Gaussian contour equation showing the original and
the translated-rotated sets of axes.
46. Consider Problem 45. Determine: (a) E ( y

− 0 .

| x = 0), (b) f y | x ( β | 0), (c) P (0 < y < 4 | x = 0),

10).
47. Assume x and y are jointly Gaussian with x

(d) P (3 < x <

∼ G (2 , 13), y ∼ G (1 , 8), and ρ = x , y

− 8835. (a) Draw a sketch of the constant contour equation for the standardized RVs

z 1 and z 2 . (b) Using the results of (a), Draw a sketch of the joint Gaussian constant
contour for x and y .

5 .