Home Assignment №7
DSBA Statistics 2025–26
Continuous Random Variables. Student’s, χ2 distributions
Dateline: 23:59, Monday, November 17, 2025
1. (6 points) Let X equal the number of telephone calls that are received on-campus college
switchboard during a 15-minute period. The following numbers of calls were received
during each of 26 time periods:
4 8 5 3 1 3 2 5 6 7 4 4 5
2 3 6 4 1 2 5 6 7 5 7 5 1
(a) Assume that the parameter of a Poisson distribution is λ = 4.2. Compare P(X ≤ 3)
with the proportion of observations that are less than or equal to 3.
(b) Compare P(X > 5) with the proportion of observations that are greater than 5.
(c) What is the probability that the time between two calls will be less than 4 minutes?
Greater than 12 minutes?
(d) Does it look like the Poisson distribution with λ = 4.2 could be correct probability
model on these limited data?
2. (6 pts) In a small town, each of 500 people votes for a candidate A with probability 0.6.
Estimate the probability that the candidate A will
(a) receive 294 to 300 votes;
(b) win the election in the first round.
Find the exact value of probability in a). Is there necessity to find the exact value of
probability in b)?
3. (5 pts) Let X ∼ Exp(λ), E(X) = 1/2, Y = e−X . Find E(Y ), V(Y ).
(
α · x3 , 0 < x < 1
4. (5 pts) Let X have a pdf: fX (x) =
0,
otherwise.
(a) Find α, E(x), V(X).
(b) What is the pdf of Y = X 2 ?
5. (6 pts) Let X ∼ U (a, b). How are random variables Y1 =
tributed? When is the distribution of Y2 uniform?
X −a
and Y2 = |X| disb−a
6. (7 pts) Let X1 , X2 , X3 , X4 , X5 , X6 be independent standard normal random variables.
Find
(a) P(X12 + X22 > 1), P(X12 + X22 > 1). Are they equal? Provide the intuition.
p
p
(b) P( X12 + X22 > 2X3 ), P( X12 + X22 > X3 + X4 ).
(c) P(X12 + X22 > 2X32 ), P(X12 + X22 > X32 + X42 ).
(d) P (X1 − X2 )2 + (X3 − X4 )2 > 3 .
7. (4 pts) Suppose Xi ∼ N (0, 3), for i = 1, 2, 3, 4. Assume all these random variables are
independent. Derive the value of k in each of the following:
(a) P(X1 + 3X2 > 4) = k.
(b) P(X12 + X22 + X32 + X42 < k) = 0.99.
p
(c) P(X1 < k X22 + X32 ) = 0.05.
(d) P X12 + X22 > 39(X32 + X42 ) = k.
2