### Probability and Statistics Problem Set**Problem Q4:**The probability density function (PDF) of the random variable (RV) \( X \) is given by:\[ f(x) = 5e^{-kx} \text{ for } x \geq 0. \]Tasks:a) Determine the value of \( k \).b) Calculate the probability that \( X > 1 \).c) Given \( Y \sim N(2, 2) \) and \( X \) and \( Y \) are independent, find the probability:\[ P(0.25 \leq X \leq 0.5 \text{ and } 0.1 \leq Y \leq 0). \]**Notes:**- Utilize integration to solve part (a) for \( k \) to ensure the total area under the PDF is equal to 1.- For part (b), use the cumulative distribution function (CDF) or direct integration.- To solve part (c), calculate the probabilities separately for independent events, using properties of the normal distribution for \( Y \).

The pdf of X is f(x)=5e-kx for x≥0.

Q4. The probability density function of the RV X is f(x) = 5e-kx for x 20. Find 2) bslubsriba ni Hit (di a) the value of k. tions: b) the probability that x>1. 0.95 c) If Y~N(2,2). X and Y are independent, then find the P(0.25 ≤ x ≤ 0.5 and 0.1 ≤ y ≤ 0). ods art ni iau bulev 19 (otsv 15/4 apolov Isisini novi faci prli vino be

Q4. The probability density function of the RV X is f(x) = 5e-kx for x 20. Find 2) bslubsriba ni Hit (di a) the value of k. tions: b) the probability that x>1. 0.95 c) If Y~N(2,2). X and Y are independent, then find the P(0.25 ≤ x ≤ 0.5 and 0.1 ≤ y ≤ 0). ods art ni iau bulev 19 (otsv 15/4 apolov Isisini novi faci prli vino be

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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