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    Assignment 3

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    Assignment 3

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    Assignment 3

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    DS303: Statistical Foundations of Data Science

    Assignment No: 3

    Course Instructor: Siddhartha Sarma Date: 14 Mar 2024


    Submission deadline: 4 Apr 2024, 11 AM (Bring hardcopies to the classroom)

    Related topics:
    ˆ Continuous random variables and their distributions
    ˆ Functions of a random variable

    Notations:

    ˆ FX (.), fX (.) denote cumulative distribution function(CDF) and probability den-


    sity function (pdf) of a random variable X, respectively.

    ˆ E[X], Var(X) denote the expectation and variance of random variable X, re-
    spectively.

    ˆ The characteristic function (also known as Moment generating function) of a


    random variable X is defined as ΦX (u) = E[ejuX ].

    ˆ ‘∼’ represents distributed as.

    ˆ Exp(λ) denotes exponential random variable with parameter λ, i.e, fX (x) =


    λe−λx , x ≥ 0.

    ˆ Unif(a, b) denotes continuous uniform distribution with a and b as left and right
    limits, respectively.

    ˆ N (µ, σ) denotes Gaussian


    √ distribution
    R ∞ −u2 /2 with mean µ and standard deviation
    σ > 0 and Q(x) = (1/ 2π) x e du

    Note: For the questions prefixed with [PA], in addition to solving them analytically, you should write
    a program in any programming language of your choice to generate probability density functions
    numerically and compare them to their corresponding probabilities calculated analytically. Please
    refer to the discussion section of the course moodle page for examples.
    A separate submission link for the programming assignments will be provided on the course
    moodle page.

    (
    U − 0.5, U ≥ 0.5
    1. [PA] Let X = , where U ∼ Unif(0, 1)
    0, U < 0.5
    (a) Find and carefully sketch the CDF FX (.). In particular, what is FX (0)?
    (b) Find the characteristic function ΦX (u).

    1
    2. [PA] Let X be exponentially distributed with mean 1/λ. Find and carefully
    sketch the distribution functions for the random variables Y = exp(X) and
    Z = min{X, 3}.
    3. If a random variable X has the following CDF,


     0, x<0

    2
    2x /3, 0 ≤ x < 0.5



    FX (x) = 1 − exp(−0.75x) 0.5 ≤ x < 1.5

    x/4 − 0.05, 1.5 ≤ x < 2





    1, x≥2

    (a) Find Pr(X = i), i = 0.5, 1, 1.5, 2


    (b) Find Pr( 12 < X < 32 ).

    [PA] Write a program that generates samples from the above distribution and
    verify its output by plotting a histogram of the cumulative function of the
    samples generated by your program.
    4. [PA] If X ∼ Unif(−2π, 2π), then find fY (y) if (a) Y = X 3 , (b) Y = X 4 , and
    (c) Y = 2 sin(3X + 40◦ ).
    5. [PA] Let FX (x) be the CDF of a random variable X. Find the distribution
    (CDF) of random variable Y = FX (X). Crosscheck you answer for the following:
    X ∼ Exp(λ), X ∼ N (µ, σ).
    6. [PA] We place at random 200 points in the interval (0, 100). The distance from
    0 to the first random point is a random variable Z. Find FZ (z).
    7. Find the mean and variance of random variables with the following characteristic
    functions: (a) Φ(u) = exp(−5u2 + 2ju), (b) Φ(u) = (exp(ju) − 1)/ju, and (c)
    Φ(u) = exp(λ(exp(ju) − 1)).
    8. Show that for a continuous random variable X (recall Q.2 of Assignment#2 )
    Z ∞ Z 0
    E[X] = (1 − FX (x))dx − FX (x)dx
    0 −∞

    9. Express each of the given probabilities in terms of the standard Gaussian com-
    plementary CDF Q(.) (a) Pr(X ≥ 16), (b) Pr(X 2 ≥ 16), where X ∼ N (10, 3).

    Some questions were taken from the following books


    ˆ Papoulis & Pillai, “Probability, Random Variables and Stochastic Processes”, McGraw Hill,
    4th Ed
    ˆ Sheldon Ross, “A first course in Probability”, 8th Ed
    ˆ Lecture notes of Prof. Hajek, University of Illinois Urbana-Champaign

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