Convolution of Discrete Random Variables
Let X and Y be independent discrete random variables with the following probability mass functions: For X: P(X = 0) = 0.3, P(X = 1) = 0.5, P(X = 2) = 0.2. For Y: P(Y = 0) = 0.6, P(Y = 1) = 0.4. (a) Construct the probability mass function of S = X + […]
Part a
Using the provided pmfs for X and Y, construct the pmf of S = X + Y via discrete convolution. Show all steps detailing the combination of outcomes.
Calculation of Expected Value from PMF
Given a discrete random variable X with probability mass function (pmf) defined as follows: \(P(X=x)=\begin{cases}0.1, &\text{for } x=1\\0.3, &\text{for } x=2\\0.4, &\text{for } x=3\\0.2, &\text{for } x=4\end{cases}\), calculate the expected value \(E(X)\).
Binomial Probability Calculations
A factory produces light bulbs, and it is known that 95% of these bulbs function properly. In a batch of 10 randomly selected bulbs: (a) Calculate the probability that none of the light bulbs are defective, denoted as \(P(X=0)\). (b) Calculate the probability that exactly one light bulb is defective, denoted as \(P(X=1)\). (c) Discuss […]
Definition and Properties of Discrete Random Variables
Define a discrete random variable and state the two essential conditions that its probability mass function (PMF) must satisfy.
Multi-step Poisson Distribution Probability Calculations
Consider a call center where calls arrive following a Poisson process with an average rate of 4 calls per hour. (a) Calculate the probability of receiving exactly 3 calls in one hour. (b) Calculate the probability of receiving between 2 and 5 calls (inclusive) in one hour.
Poisson Probability Computation
A random variable \(X\) follows a Poisson distribution with a mean of \(\lambda = 4\). Compute \(P(X = 3)\) using the probability mass function of the Poisson distribution, where \(P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}\).
Variance Transformation of a Discrete Random Variable
Let X be a discrete random variable with the following probability mass function: – P(X = 0) = 0.25 – P(X = 1) = 0.50 – P(X = 2) = 0.25 Using the rules for linear transformations, calculate Var(4X – 3).
Binomial Distribution Probability Calculation
In a manufacturing process, 40% of items produced are defective. In a random sample of 6 items, calculate the probability of finding exactly 3 defective items, assuming each item’s quality is independent.
Computing the Binomial Probability for k = 2
A random variable \(X\) follows a binomial distribution with parameters \(n = 5\) and \(p = 0.3\). Compute \(P(X = 2)\) using the binomial probability mass function.
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