Practice Examples (11.5) - Moments and Moment Generating Functions - Mathematics - iii (Differential Calculus) - Vol 3

Practice: Examples

Lesson 11.5Moments and Moment Generating Functions

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9 questions

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3 questions · about 2 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
What is the MGF of a discrete random variable taking values 0 and 1 with equal probability?
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Use the definition of MGF, E[e^(tX)].
Question 2.
Calculate the mean of a discrete random variable that takes on values 1 and 3 with probabilities 1/4 and 3/4 respectively.
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Use the formula E[X] = Σ(x * P(X=x)).
3Question 3.
What is the MGF of a discrete random variable with possible values 0 and 1 each with probability 1/2?
- (1 + e^t)/2
- (1 + e^(t/2))/2
- e^t
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Recall the formula for the MGF.
4Question 4.
True or False: The mean of a normally distributed random variable is always zero.
- True
- False
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Consider the definition of a normal distribution.
Question 5.
A discrete random variable X takes values 2, 3, and 5 with probabilities 0.3, 0.5, and 0.2 respectively. Calculate the MGF and derive the mean and variance.
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Use the definition of MGF and the formulas for mean and variance.
Question 6.
Assume Y ~ N(μ, σ²). Prove that the mean can be derived from the MGF.
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Differentiate the MGF and evaluate at t=0.

5 more questions in this section. Enroll free to practise them all.