This set of Probability and Statistics Multiple Choice Questions & Answers (MCQs) focuses on “Mathematical Expectation”.

1. The expectation of a random variable X(continuous or discrete) is given by _________

a) ∑xf(x), ∫xf(x)

b) ∑x^{2} f(x), ∫x^{2} f(x)

c) ∑f(x), ∫f(x)

d) ∑xf(x^{2}), ∫xf(x^{2})

View Answer

Explanation: The expectation of a random variable X is given by the summation (integral) of x times the function in its interval. If it is a continuous random variable, then summation is used and if it is discrete random variable, then integral is used.

2. Mean of a random variable X is given by _________

a) E(X)

b) E(X^{2})

c) E(X^{2}) – (E(X))^{2}

d) (E(X))^{2}

View Answer

Explanation: Mean is defined as the sum of the function in its domain multiplied with the random variable’s value. Hence mean is given by E(X) where X is a random variable.

3. Variance of a random variable X is given by _________

a) E(X)

b) E(X^{2})

c) E(X^{2}) – (E(X))^{2}

d) (E(X))^{2}

View Answer

Explanation: Variance of a random variable is nothing but the expectation of the square of the random variable subtracted by the expectation of X (mean of X) to the power 2. Therefore the variance is given by E(X

^{2}) – (E(X))

^{2}.

4. Mean of a constant ‘a’ is ___________

a) 0

b) a

c) a/2

d) 1

View Answer

Explanation: Let f(x) be the pdf of the random variable X.

Now, E(a) = ∫af(x)

= a∫f(x)

= a(1) = a.

5. Variance of a constant ‘a’ is _________

a) 0

b) a

c) a/2

d) 1

View Answer

Explanation: V(a) = E(a

^{2}) – (E(X))

^{2}

= a

^{2}– a

^{2}

= 0.

6. Find the mean and variance of X?

x | 0 | 1 | 2 | 3 | 4 |

f(x) | 1/9 | 2/9 | 3/9 | 2/9 | 1/9 |

a) 2, 4/3

b) 3, 4/3

c) 2, 2/3

d) 3, 2/3

View Answer

Explanation: Mean = \(E(X) = ∑f(x) = 0(\frac{1}{9}) + 1(\frac{2}{9}) + 2(\frac{3}{9}) + 3(\frac{2}{9}) + 4(1/9) \)

= 2

Variance \( = E(X^2)-(E(X))^2 = (0 + \frac{2}{9} + \frac{12}{9} + \frac{28}{9} + \frac{26}{9}) – 4 \)

\( = \frac{4}{3} \).

7. Find the expectation of a random variable X?

x | 0 | 1 | 2 | 3 |

f(x) | 1/6 | 2/6 | 2/6 | 1/6 |

a) 0.5

b) 1.5

c) 2.5

d) 3.5

View Answer

Explanation: \(E(X) = 0(\frac{1}{6}) + 1(\frac{2}{6}) + 2(\frac{2}{6}) + 3(\frac{1}{6}) = 1.5. \)

8. 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

View Answer

Explanation: \(\int_0^∞ ke^{-x} dx = 1 \)

kГ(1) = 1

k = 1

Now, \(E(X) = \int_0^∞ xe^{-x} dx = Г(2) = 1.\)

9. 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

View Answer

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.

10. Find the mean of a continuous random variable X if f(x) = 2e^{-x} for x>0 and -e^{x} for x<0.

a) 0

b) 1

c) 2

d) 3

View Answer

Explanation: \(E(X) = \int_0^∞ 2xe^{-x} dx + \int_{-∞}^0 xe^x dx \)

= 2 Г(2) + Г(2) = 3.

11. What is moment generating function?

a) M_{x}(t) = E(e^{tx})

b) M_{x}(t) = E(e^{-tx})

c) M_{x}(t) = E(e^{2tx})

d) M_{x}(t) = E(e^{t})

View Answer

Explanation: Moment generating function is nothing but the expectation of e

^{tX}. So, the function is multiplied with e

^{tX}before performing the integration or summation.

12. 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) \((\frac{e^{2t}-1}{t^2}) \)

View Answer

Explanation: M

_{x}(t) = E(e

^{tx}) = \(\int_0^1 xe^{tx} dx+\int_1^2 (2-x) e^{tx} dx + 0 = (\frac{e^t-1}{t})^2. \)

13. E(X) = npq is for which distribution?

a) Bernoulli’s

b) Binomial

c) Poisson’s

d) Normal

View Answer

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.

14. E(X) = λ is for which distribution?

a) Bernoulli’s

b) Binomial

c) Poisson’s

d) Normal

View Answer

Explanation: In Poisson’s distribution, there is a positive constant λ which is the mean of the distribution and variance of the distribution.

15. E(X) = μ and V(X) = σ^{2} is for which distribution?

a) Bernoulli’s

b) Binomial

c) Poisson’s

d) Normal

View Answer

Explanation: In Normal distribution, the mean and variance is given by μ and σ

^{2}respectively. In case of standard normal distribution the mean is 0 and the variance is 1.

**Sanfoundry Global Education & Learning Series – Probability and Statistics.**

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