Exam

Probability and Statistics

Bernardo Guimaraes

September 2007

1. (9 pts) A researcher collected 100 observations on a random variable X and calculated the mean

and the median. He obtained M ean(X) = 10 and M edian(X) = 8. Subsequently, he realized

that the highest observation in his sample was misreported. He had x1 = 20, but the correct value

was x1 = 21.

(a) Using the correct value of x1 , what is the mean of X? If it can’t be calculated, explain why.

(b) Using the correct value of x1 , what is the median of X? If it can’t be calculated, explain

why.

(c) What do the values of M ean(X) and M edian(X) hint about asymmetry?

2. (7 pts) Let X and Y be random variables with joint probability density function

fX,Y (x, y) = 2

for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and 0 ≤ x + y ≤ 1. Are X and Y independent? Explain.

3. (11 pts) X and Y are independent random variables, with marginal density functions:

x

, 0≤x≤2

2

fY (y) = 1 , 0 ≤ y ≤ 1

fX (x) =

Find the joint cumulative distribution function for all possible values of x and y.

4. (9 pts) A supermarket has 2 express lines. Let X and Y denote the number of customers in the

first and the second, respectively, at a given time. Denote by pX,Y (x, y) the joint probability

function of X and Y .

(a) Would you expect X and Y to be independent?

(b) Would you expect pX,Y (x, y) to be symmetric, that is, would you expect pX,Y (a, b) =

pX,Y (b, a)?

(c) Would you expect pX,Y (5, 2) to be bigger than pX,Y (2, 2)? Or bigger than pX,Y (5, 5)?

5. (7 pts) Let X and Y be random variables such that Y = 2X + 2. The probability density function

of X is:

fX (x) = 1 , 0 ≤ x ≤ 1

What is the probability density function of Y ?

6. (10 pts) Consider the random variable Y , such that Pr(Y = 1) = θ, and Pr(Y = 0) = 1 − θ.

A sample consisting of n independent realizations of Y ({y1 , y2 , ..., yn }) was collected. Find the

method of moments estimator of θ.

1

7. (11 pts) Consider the random variable X, such that Pr(X = x) = (1 − p)x−1 p, for x = 1, 2, ....

A sample consisting of n independent realizations of X ({x1 , x2 , ..., xn }) was collected. Find the

maximum likelihood estimator of p.

8. (11 pts) The time between eruptions of a given volcano, T , follows an exponential distribution,

with hazard rate λ (see formulas below). A researcher wants to test whether λ < 5, with level of

significance 5%. For that, he has only one observation, t.

(a) What are the null and alternative hypotheses?

(b) What are the values of t that lead the researcher to reject the null hypothesis?

9. (12 pts) X is a random variable described by an exponential distribution.

A researcher collected a sample consisting of 81 independent realizations of X ({x1 , x2 , ..., x81 }).

She found that:

81

81

X

X

xi = 6480 and

(xi − x

¯)2 = 6480

i=1

i=1

where x

¯ is the sample mean. To simplify calculations, note that 6480 = 80 × 81.

Find a confidence interval for the mean of the distribution (µ) with 95% confidence.

10. (13 pts) Consider a normal random variable with unknown mean µ and variance σ 2 = 4. A

researcher wants to test whether µ is greater than 0.

(a) What are the null and alternative hypotheses?

(b) At 5% level of significance, what should be the size of the sample if the researcher wants to

reject the null hypothesis with probability 90% if µ = 1?

__________________________________

• You can apply the Central Limit Theorem for n > 30.

√

√

√

• For calculations, you can use: 2 = 1.4, 3 = 1.7 and 5 = 2.2.

• Exponential distribution (hazard rate λ):

— Probability density function: f (t) = λe−λt .

— Cumulative distribution function: f (t) = 1 − e−λt .

— Expected value: µ =

1

λ

• Normal distribution:

— Probability density function: f (x) =

1 x−µ

√ 1 e− 2 ( σ )

2πσ

2

2

.

Probability and Statistics

Bernardo Guimaraes

September 2007

1. (9 pts) A researcher collected 100 observations on a random variable X and calculated the mean

and the median. He obtained M ean(X) = 10 and M edian(X) = 8. Subsequently, he realized

that the highest observation in his sample was misreported. He had x1 = 20, but the correct value

was x1 = 21.

(a) Using the correct value of x1 , what is the mean of X? If it can’t be calculated, explain why.

(b) Using the correct value of x1 , what is the median of X? If it can’t be calculated, explain

why.

(c) What do the values of M ean(X) and M edian(X) hint about asymmetry?

2. (7 pts) Let X and Y be random variables with joint probability density function

fX,Y (x, y) = 2

for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and 0 ≤ x + y ≤ 1. Are X and Y independent? Explain.

3. (11 pts) X and Y are independent random variables, with marginal density functions:

x

, 0≤x≤2

2

fY (y) = 1 , 0 ≤ y ≤ 1

fX (x) =

Find the joint cumulative distribution function for all possible values of x and y.

4. (9 pts) A supermarket has 2 express lines. Let X and Y denote the number of customers in the

first and the second, respectively, at a given time. Denote by pX,Y (x, y) the joint probability

function of X and Y .

(a) Would you expect X and Y to be independent?

(b) Would you expect pX,Y (x, y) to be symmetric, that is, would you expect pX,Y (a, b) =

pX,Y (b, a)?

(c) Would you expect pX,Y (5, 2) to be bigger than pX,Y (2, 2)? Or bigger than pX,Y (5, 5)?

5. (7 pts) Let X and Y be random variables such that Y = 2X + 2. The probability density function

of X is:

fX (x) = 1 , 0 ≤ x ≤ 1

What is the probability density function of Y ?

6. (10 pts) Consider the random variable Y , such that Pr(Y = 1) = θ, and Pr(Y = 0) = 1 − θ.

A sample consisting of n independent realizations of Y ({y1 , y2 , ..., yn }) was collected. Find the

method of moments estimator of θ.

1

7. (11 pts) Consider the random variable X, such that Pr(X = x) = (1 − p)x−1 p, for x = 1, 2, ....

A sample consisting of n independent realizations of X ({x1 , x2 , ..., xn }) was collected. Find the

maximum likelihood estimator of p.

8. (11 pts) The time between eruptions of a given volcano, T , follows an exponential distribution,

with hazard rate λ (see formulas below). A researcher wants to test whether λ < 5, with level of

significance 5%. For that, he has only one observation, t.

(a) What are the null and alternative hypotheses?

(b) What are the values of t that lead the researcher to reject the null hypothesis?

9. (12 pts) X is a random variable described by an exponential distribution.

A researcher collected a sample consisting of 81 independent realizations of X ({x1 , x2 , ..., x81 }).

She found that:

81

81

X

X

xi = 6480 and

(xi − x

¯)2 = 6480

i=1

i=1

where x

¯ is the sample mean. To simplify calculations, note that 6480 = 80 × 81.

Find a confidence interval for the mean of the distribution (µ) with 95% confidence.

10. (13 pts) Consider a normal random variable with unknown mean µ and variance σ 2 = 4. A

researcher wants to test whether µ is greater than 0.

(a) What are the null and alternative hypotheses?

(b) At 5% level of significance, what should be the size of the sample if the researcher wants to

reject the null hypothesis with probability 90% if µ = 1?

__________________________________

• You can apply the Central Limit Theorem for n > 30.

√

√

√

• For calculations, you can use: 2 = 1.4, 3 = 1.7 and 5 = 2.2.

• Exponential distribution (hazard rate λ):

— Probability density function: f (t) = λe−λt .

— Cumulative distribution function: f (t) = 1 − e−λt .

— Expected value: µ =

1

λ

• Normal distribution:

— Probability density function: f (x) =

1 x−µ

√ 1 e− 2 ( σ )

2πσ

2

2

.

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