ECON 120B Midterm Exam, Summer-I 2023
1. (4 points) For the following claims, determine if they are true or false. Write T or F in the boxes. You do not need to provide any explanation.
(a) Assume X follows a normal distribution with mean µ < 0 and variance σ2 > 0. Then X2 also follows a normal distribution.
(b) If X and Y are independent random variables, then they have zero correlation.
(c) We can always find the marginal distributions of two random variables, X and Y , from their joint distribution.
(d) Under the null hypothesis, the size of a test is 0.05.
2. (6 points) Show each step in your calculation/derivation. Assume X ∼ Bernoulli(0.3).
(a) Find the expectation of 3X − 1 (1 point).
(b) Find the standard deviation of −1(X + 2) (1 point).
(c) Write down the distribution of (X − 1)2 , and find its expectation. (1 + 1 = 2 points)
(d) Let Y ∼ Bernoulli(0.1). Assume X and Y are independent. Find r[X + Y = 1]. (2 points)
3. (4 points) Show each step in your calculation/derivation. The following table gives the joint probability distribution between X and Y.
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Y = 1
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Y = 2
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Y = 4
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total
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X = 0
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0.10
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0.05
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0.30
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0.45
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X = 1
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0.20
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0.20
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0.15
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0.55
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total
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0.30
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0.25
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0.45
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1.00
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(a) Find the expectation E[Y]. (1 point)
(b) Find the conditional distribution of X given Y = 2. (1 point)
(c) Find the distribution of X × Y. (2 points)
4. (4 points) Let X1 , X2 ,..., Xn be a random sample from the distribution Ⅵ(µ,4). A researcher is interested in testing the hypothesis H0 : µ = 2 vs. H1 : µ 2. She decides to use the following test:
Reject H0 if |X1 − 2| > 1.
In other words, the researcher only looks at the first observation.
(a) Find the size of the her test. Show each step in your calculation/derivation. (2 points)
(b) Assume in the dataset we have X1 = −2. Would the researcher reject the null hypothesis? Just write Yes or No. (1 point)
(c) Do you like the above test? Briefly explain your reasoning. (1 point)
5. (3 points) Let Y1 , Y2 ,..., Yn be a random sample from a distribution with mean μ . A test of H0 : μ = 10 vs. H1 : μ 10 yields a p-value of 0.07.
(a) Does the 90% confidence interval contain μ = 10? Explain with one or two sentences. (1.5 points)
(b) Can you determine if μ = 8 is contained in the 95% confidence interval? Explain with one or two sentences. (1.5 points)
6. (5 points) Consider the following Stata output. Find (a), (b), (c), (d), (e). Show each step in your calculation/derivation.
. ttest X == 14
One-sample t test
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Variable | Obs Mean Std . err . Std . dev . [95% conf . interval]
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X | 420 15.77048 (a) 18.28453 (b) (c)
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mean = mean(X) t = (d)
H0: mean = 14 Degrees of freedom = 419
Ha: mean < 14 Ha: mean != 14 Ha: mean > 14
Pr(T < t) = 0.9761 Pr(|T| > |t|) = (e) Pr(T > t) = 0.0239
7. (4 points) Let X be a random variable. Consider the following mathematical derivation:
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V[X]
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(a)
(b)
(c)
(d)
(e)
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El(X − E[X])2]
ElX2 − 2XE[X] +（E[X]）2]
ElX2]− El2XE[X]] + (ElX])2 ElX2]− 2 (ElX])2 + (ElX])2 ElX2]− (ElX])2 .
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Equality (a) is the definition of variance. Equalities (b) and (e) are just algebra. Explain the reasoning behind Equalities (c) and (d). (2 + 2 = 4 points)