1. A mobile conversation follows a exponential distribution f (x) = (1/3)e-x/3. What is the probability that the conversation takes more than 5 minutes?
a) e-5/3
b) e-15
c) 5e-15
d) e-5/3
Explanation: f(x) = (1/3)e-x/3. The call should last more than 5 minutes so integrating from 5 till infinity we get
\(\frac{1}{3} ∫ (e^{-x/3}dx) = \frac{1}{3}(\frac{- e^{-5/3}}{-1/3}) \)
= e-5/3.
2. Exponential distribution is bi-variate.
a) True
b) False
Explanation: Exponential distribution is uni-variate.
It is only defined for non-negative variables.
3. A random variable X has an exponential distribution with probability distribution function is given by
f(x)= 3e-3x for x>0 = 0 otherwise
Find probability that X is not less than 2.
a) e-3
b) e-6 – 3
c) e-6
d) e-6 – 1
Explanation: Probability of the function taking values from 2 to infinity.
P(X > 2) = 1 – P(X < 2) = Integrating the function from 0 to 2 we get
P(X = 1) = 1 – [3(e0 – e-6)/(-3)]
= e-6.
4. Consider a random variable with exponential distribution with λ=1. Compute the probability for P (X>3).
a) e-3
b) e-1
c) e-2
d) e-4
Explanation: The function takes values from 3 to infinity. This can be written alternatively as integrating from 0 to 3 and subtracting the whole from
P(X > 3) = 1 – P(X < 3)
= 1 – (1 – e-3)
= e-3.
5. The mean and the variance for gamma distribution are __________
a) E(X) = 1/λ, Var(X) = α/λ2
b) E(X) = α/λ, Var(X) = 1/λ2
c) E(X) = α/λ, Var(X) = α/λ2
d) E(X) = αλ, Var(X) = αλ2
Explanation: The mean and the variance for gamma distribution is given as
E(X) = α/λ, Var(X) = 1/λ2.
6. Putting α=1 in Gamma distribution results in _______
a) Exponential Distribution
b) Normal Distribution
c) Poisson Distribution
d) Binomial Distribution
Explanation: f (x) = λα xα−1 e−λx / Γ(α) for x > 0
= 0 otherwise
If we let α=1, we obtain
f(x) = λe−λx for x > 0
= 0 otherwise.
Hence we obtain Exponential Distribution.
7. Sum of n independent Exponential random variables (λ) results in __________
a) Uniform random variable
b) Binomial random variable
c) Gamma random variable
d) Normal random variable
Explanation: Gamma (1,λ) = Exponential (λ).
Hence Exponential (λ1) + Exponential (λ2) + Exponential (λ3)….. n times = Gamma(n,λ).
8. Find the value of Γ(5/2).
a) 5/4 . π1/2
b) 7/4 . π1/2
c) 1/4 . π1/2
d) 3/4 . π1/2
Explanation: By the property of Gamma Function
Γ(α+1) = αΓ(α)
\(Γ(\frac{5}{2}) = \frac{3}{2} ⋅ Γ(\frac{3}{2}) \)
\(= \frac{3}{2} ⋅ \frac{1}{2} . Γ(\frac{1}{2}) \)
\(= \frac{3}{2} ⋅ \frac{1}{2} ⋅ π^{1/2} \) By property of Gamma function \(Γ(\frac{1}{2}) = π^{1/2} \)
\(= \frac{3}{4} . π^{1/2}. \)
9. Gamma function is defined as Γ(α) = 0∫∞ xα−1 e−x dx.
a) True
b) False
Explanation: The Gamma function is defined as Γ(α) = 0∫∞ xα−1 e−x dx. Gamma function can also be defined as Γ(α+1) = αΓ(α).
10. Gamma distribution is Multi-variate distribution.
a) True
b) False
Explanation: Gamma distribution is a uni-variate distribution that means it is only defined for x ranging from (0, ∞).