1. Mean of a constant ‘a’ is ___________
a) 0
b) a
c) a/2
d) 1
Explanation: Let f(x) be the pdf of the random variable X.
Now, E(a) = ∫af(x)
= a∫f(x)
= a(1) = a.
2. Variance of a constant ‘a’ is _________
a) 0
b) a
c) a/2
d) 1
Explanation: V(a) = E(a2) – (E(X))2
= a2 – a2
= 0.
3. Find the expectation of a random variable X if f(x) = ke-x for x>0 and 0 otherwise.
a) 0
b) 1
c) 2
d) 3
Explanation: \(\int_0^∞ ke^{-x} dx = 1 \)
kГ(1) = 1
k = 1
Now, \(E(X) = \int_0^∞ xe^{-x} dx = Г(2) = 1.\)
4. Find the mean of a random variable X if f(x) = x – 5⁄2 for 0<x<1 and 2x for 1<x<2 and 0 otherwise.
a) 3.5
b) 3.75
c) 2.5
d) 2.75
Explanation: \(E(X) = \int_0^1 (x-5/2)dx+∫_1^2(2x)dx+0 \)
\(= (\frac{x^3}{3} – \frac{5x^2}{4}) \) {from 0 to 1} \( + (\frac{2x^3}{3}) \) {from 1 to 2}
\(= \frac{1}{3} – \frac{5}{4} + \frac{16}{3} – \frac{2}{3} \)
= 3.75.
5. Find the mean of a continuous random variable X if f(x) = 2e-x for x>0 and -ex for x<0.
a) 0
b) 1
c) 2
d) 3
Explanation: \(E(X) = \int_0^∞ 2xe^{-x} dx + \int_{-∞}^0 xe^x dx \)
= 2 Г(2) + Г(2) = 3.
6. What is moment generating function?
a) Mx(t) = E(etx)
b) Mx(t) = E(e-tx)
c) Mx(t) = E(e2tx)
d) Mx(t) = E(et)
Explanation: Moment generating function is nothing but the expectation of etX. So, the function is multiplied with etX before performing the integration or summation.
7. Find the Moment Generating Function of f(x) = x for 0<x<1 and 2-x for 1<x<2 and 0 otherwise.
a) \((\frac{e^t-1}{t})^2 \)
b) \((\frac{e^{-t}-1}{t})^2 \)
c) \((\frac{e^{2t}-1}{t})^2 \)
d) \((\frac{e^{2t}-1}{t^2}) \)
Explanation: Mx(t) = E(etx) = \(\int_0^1 xe^{tx} dx+\int_1^2 (2-x) e^{tx} dx + 0 = (\frac{e^t-1}{t})^2. \)
8. E(X) = npq is for which distribution?
a) Bernoulli’s
b) Binomial
c) Poisson’s
d) Normal
Explanation: In binomial distribution, probability of success is given by p and that of failure is given by q and the event is done n times. The mean of this distribution is given by npq
9. E(X) = λ is for which distribution?
a) Bernoulli’s
b) Binomial
c) Poisson’s
d) Normal
Explanation: In Poisson’s distribution, there is a positive constant λ which is the mean of the distribution and variance of the distribution.
10. E(X) = μ and V(X) = σ2 is for which distribution?
a) Bernoulli’s
b) Binomial
c) Poisson’s
d) Normal
Explanation: In Normal distribution, the mean and variance is given by μ and σ2 respectively. In case of standard normal distribution the mean is 0 and the variance is 1.