Unit 3 - Practice Quiz

1 The expected value of a continuous random variable with probability density function is given by:

2 If is a random variable and is a constant, what is ?

3 Given a random variable and constants and , what is ?

4 If is a function of a continuous random variable , the expected value is calculated as:

5 The -th moment of a random variable about the origin is denoted by and is defined as:

6 The first moment about the origin, , represents the:

7 The -th central moment is defined as:

8 The second central moment is commonly known as the:

9 Which of the following relationships relates the variance , the mean square value , and the mean ?

10 What is the value of the first central moment, ?

11 For a random variable and constants , the variance is equal to:

12 Skewness is a measure of:

13 The coefficient of skewness is defined using the third central moment and standard deviation as:

14 Kurtosis relates to which central moment?

15 If a PDF is symmetric about its mean, then all odd order central moments are:

16 Chebyshev's Inequality provides a bound on probability for:

17 According to Chebyshev's Inequality, is less than or equal to:

18 Using Chebyshev's inequality, what is the lower bound for the probability that lies within $2$ standard deviations of the mean ()?

19 The Characteristic Function of a random variable is defined as:

20 The value of the characteristic function at the origin, , is always:

21 The maximum absolute value of the characteristic function is:

22 If is the characteristic function of , the -th moment of can be found by:

23 The Moment Generating Function (MGF) is defined as:

24 Using the MGF , the -th moment is calculated as:

25 Unlike the characteristic function, the moment generating function:

26 If , and is the MGF of , then is:

27 The characteristic function of the sum of two independent random variables is:

28 Given a transformation , if is a monotonic differentiable function, the PDF of , , is given by:

29 If and is a continuous random variable, then is:

30 Consider the transformation . If the PDF of is , what is for ?

31 For a linear transformation , the mean is:

32 For a linear transformation , the standard deviation is:

33 If the characteristic function of is , then has which distribution?

34 The second derivative of the Moment Generating Function at yields:

35 Which inequality states that for a non-negative random variable , ?

36 If is a discrete random variable with PMF , the expected value is:

37 If the variance of is zero, then:

38 If , how are the characteristic functions related?

39 The Maclaurin series expansion of the Characteristic Function generates coefficients related to:

40 Let be a random variable with mean and variance . Using Chebyshev's inequality, is at most:

41 The characteristic function of a standard normal random variable is:

42 Which property allows us to recover the PDF from the Characteristic Function?

43 For a random variable , the quantity is called:

44 If for , what is ?

45 The skewness of the normal distribution is:

46 If is Uniformly distributed in , then follows which distribution?

47 The cumulant generating function is defined as:

48 What is the relation between the second cumulant and the variance?

49 If exists, then:

50 For the transformation , where is constant, the PDF is: