An insect is laying 2000 eggs. A viable offspring develops from each egg (mutually independently
from all other eggs) with a probability of 0.001. Determine the probability of having at least two
viable offsprings first exactly and then compute an approximation using the Poisson distribution.
Reflect on the resultsConsider the following random experiment: We toss a biased coin with “tails”
Consider the following random experiment: We toss a biased coin with “tails”
probability p repeatedly until the first “tails” appears. The outcome ω is the number of coin flips
in this sequence before (i.e. not including) the first “tails”. Define the probability space for this
random experimentprobability p repeatedly until the first “tails” appears. The outcome ω is the number of coin flips
in this sequence before (i.e. not including) the first “tails”. Define the probability space for this
random experimentConsider the following random experiment: We toss a biased coin with “tails”
probability p repeatedly until the first “tails” appears. The outcome ω is the number of coin flips
in this sequence before (i.e. not including) the first “tails”. Define the probability space for this
random experimentConsider the following random experiment: We toss a biased coin with “tails”
probability p repeatedly until the first “tails” appears. The outcome ω is the number of coin flips
in this sequence before (i.e. not including) the first “tails”. Define the probability space for this
random experiment
Department of Mathematical Sciences, Stevens Institute of Technology Dr. Kathrin Smetana Midterm Examination MA 540 Introduction to Probability Theory Fall 2021 — Exam All answers must be motivated and clearly formulated. Explain each step in your solution. Your solutions should make very clear to the instructor that you understand all of the steps and the logic behind the steps. Please indicate clearly all the questions you solve and avoid writing your solution on different, disjoint pages. You may use a calculator for Tasks 2 and 4 only. Task 1 (Probability space) Bonus task: Consider the following random experiment: We toss a biased coin with “tails” probability p repeatedly until the first “tails” appears. The outcome ω is the number of coin flips in this sequence before (i.e. not including) the first “tails”. Define the probability space for this random experiment. (bonus points) Task 2 (Conditional Probability) Approximately 7% of the male and 5% of the female population in the US read the NY Times daily.1 About 51% of the US population is female and 49% is male. a) How large is the probability that a person selected at random from the US population reads the NY Times daily? b) Assume that a person in the US reads the NY Times daily. Calculate the probability that this person is female. Task 3 (Definition of a random variable) Compare the definition of a random variable that you got to know in this course with the following definition: “A random variable is a function from the sample space Ω to the set R of all real numbers.” Task 4 (Modeling with random variables) An insect is laying 2000 eggs. A viable offspring develops from each egg (mutually independently from all other eggs) with a probability of 0.001. Determine the probability of having at least two viable offsprings first exactly and then compute an approximation using the Poisson distribution. Reflect on the results. 1Numbers are made up. Task 5 (Independent random variables) Let X and Y be independent random variables. Let f, g : R→ R be Borel-measurable functions, meaning that f−1(A) ∈ B(R), g−1(C) ∈ B(R) for any A,C ∈ B(R). Show that the random variables f(X) and g(Y ) are independent. Task 6 Let X be a random variable with density fX(t) = βt(t− 2 3 )1[−α+1,α+1] with α, β ∈ R, α > 0. (a) Find β in terms of α. (b) Find the cumulative distribution function of X. (c) Calculate P(−α < x ≤ 1). (d) consider g : r→ r defined as g : x 7→ x3, i.e. g(x) = x3. what is the probability density function of the random variable y := g(x)? x="" ≤="" 1).="" (d)="" consider="" g="" :="" r→="" r="" defined="" as="" g="" :="" x="" 7→="" x3,="" i.e.="" g(x)="x3." what="" is="" the="" probability="" density="" function="" of="" the="" random="" variable="" y="" :="">