STAT2911
Probability and Statistical Models
(Practice) Quiz 2, 2021
NAME:
STUDENT ID:
Clearly circle the correct answer. Each problem is worth 20 marks.
1. If X1, . . . , Xn are iid with density f(x) = λ exp(−λx) for x ≥ 0, then the expectation of the
first order statistic, EX(1) is given by
(i) nλ
(ii) 1/(nλ)
(iii) λ
(iv) λ/n
(v) None of the other statements is correct
2. A random point is selected in the unit disk by choosing the radius, R, uniformly in [0, 1] and
independently choosing the angle Θ uniformly in [0, 2pi). Let X = R cos Θ and Y = R sin Θ.
The joint density of (X, Y ) is 0 except in the unit disk where it is given by
(i) 1
(ii) 1/pi2
(iii) pi2
(iv) 1/(2pi) · (x2 + y2)−1/2
(v) None of the other statements is correct
3. Let X be a binomial (n, p) RV and let Y be the number of successes in the first m ≤ n of these
independent Bernoulli trials. V (X|Y ) is:
(i) (n−m)p(1− p)
(ii) np(1− p)
(iii) mp(1− p)
(iv)
√
np(1− p)
(v) None of the other statements is correct
4. Let X be a discrete uniform RV on 0, 1, · · · ,m; that is, P (X = k) = 1/(m + 1) for k =
0, 1, . . . ,m. Let m˜ be moment estimator of m, and mˆ its MLE, given a sample x1, . . . , xn.
(i) m˜ = 2x¯ and mˆ = min{xi}
(ii) m˜ = max{xi} and mˆ = 2 min{xi}
(iii) m˜ = 1/x¯ and mˆ = 2x¯
(iv) m˜ = 2x¯ and mˆ = max{xi}
(v) None of the other statements is correct
5. If X and Y are jointly distributed with density fX,Y (x, y) = k(x− y)2, 0 < y < x < 1, then
the conditional density of X given Y = y, fX|Y (x|y), in 0 < y < x < 1, is
(i) None of the other statements is correct
(ii) 4(x−y)
2
y3
(iii) 3(x−y)
2
(1−y)3
(iv) 3(x−y)
2
(1−y)2
(v) 3(x−y)
2
x3
The following might be useful
1. The pmf of a Poisson(λ) RV is given by pX(k) = e
−λλk/k! for k = 0, 1, . . .
2. The variance of a binomial(n, p) RV is np(1− p).
3. The pmf of a hypergeometric RV X, which counts the number of red balls in a sample of m
balls taken from an urn with n balls of which r are red is given by pX(k) =
(
r
k
)(
n−r
m−k
)
/
(
n
m
)
for
k = 0, 1, . . . , r.
4. The pmf of a negative binomial(r, p) RV is given by pX(k) =
(
k−1
r−1
)
pr(1 − p)k−r, for k =
r, r + 1, . . . .
5. The mean of a geometric(p) RV is 1/p and the variance is (1− p)/p2.
6. The mean of a discrete uniform distribution on {0, 1, 2, . . . ,m} (m+ 1 possible values) is m/2
and the variance is m(m+ 2)/12.
7. The Gamma function, defined as, Γ(x) =
∫∞
0
tx−1e−t dt, satisfies Γ(α) = (α− 1)Γ(α− 1).
8. The Gamma(α, λ) distribution with shape α > 0, and rate λ > 0, is given by the pdf:
f(x) =
λα
Γ(α)
xα−1e−λx · 1x≥0.
The mean of the Gamma distribution is α/λ and its MGF is
(
λ
λ−t
)α
for t < λ.
9. The beta distribution with parameters α, β > 0 is specified by the density
f(x) =
Γ(α + β)
Γ(α)Γ(β)
xα−1(1− x)β−1 · 1[0,1](x).
10. The MGF of the N(0, 1) distribution is given by
M(t) = et
2/2 ∀t ∈ R.
11. If X, Y are jointly distributed and X ∈ L2 then V (X) = V [E(X|Y )] + E[V [X|Y ]].
12. If N and Xi are independent L
2 RVs and Xi are iid then T =
∑N
i=1Xi ∈ L2 and
V (T ) = [E(Xi)]
2 V (N) + V (Xi)E(N).
13. The t-distribution with ν degrees of freedom, t(ν), is defined as the distribution of Z/
√
S2/ν
where Z ∼ N(0, 1) is independent of S2 ∼ χ2(ν).
14. If Xi are independent N(µ, σ
2) and S2 = 1
n−1
∑n
i=1
(
Xi − X¯
)2
then
√
n
(
X¯ − µ) /S ∼ t(n− 1).
15. The density of the Cauchy distribution is given by
f(x) =
1
pi
1
1 + x2
∀x ∈ R.
16. If (X, Y ) have a joint pdf fXY and (U, V ) = T (X, Y ) where T : R2 7→ R2 is differentiable and
invertible then
fUV (u, v) = fXY (T
−1(u, v))|JT−1|.
17. A second order Taylor expansion of a real function g about the point x0 is given by
g(x) = g(x0) + g
′(x0)(x− x0) + g′′(x0)(x− x0)
2
2
+R2(x),
where limx→x0 R2(x)/(x−x0)2 = 0. In particular ex = 1+x+x2/2+R2(x), where limx→0R2(x)/x2 =
0.
18. If F is the CDF of a RV X and F is a continuous function which is differentiable everywhere
except for possibly a finite set of points, then X is a continuous RV and its density is given by
f(x) = F ′(x) where the derivative exists.
19. If F is the CDF of a continuous RV X with density f then for any x where f(x) is continuous,
F ′(x) = f(x).
20. If the CDF F has density f then the density of the first order statistic of a sample of size n
from the corresponding distribution is given by nf(x) [1− F (x)]n−1.



























































































































































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