Recursos para tareas escolares en Nigeria

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1. a. Explain the lack of memory property of exponential distribution b. The time between the arrival of an electronic message at a computer is exponentially distributed with mean of two hours. i. what is the probability that a message is received during a two-hour period? ii. If a message has not been received in the last four hours, what is the probability a message will be received in the next two-hour period? iii. Comment on the results obtained in (i) and (ii) above 2. a. State and prove Chebyshev's inequality for a random variable \( X \) b. \( X \) is a random variable with an unknown distribution with mean \( \mu_{x}=100 \) and variance \( \sigma_{x}^{2}=10 \). Use Chebyshev's inequality to estimate i. \( P(X \geq 120) \) ii. \( P(X \leq 75) \) 3. a. Show that \( \binom{n+2}{r+1}=\binom{n+1}{r+1}+\binom{n+1}{r} \) bi. Find the number n of distinct permutations that can be formed from all the letters of the word SOCIOLOGICAL bii. Find the number of ways a 4 -member committee with 2 men and 2 women can be selected from a class of 6 men and 4 women. 4. ai. Show that the MGF of a random variable \( X \) having Poisson distribution with pdf \( f(x)=\frac{e^{-t} \lambda^{x}}{x!}, x=0,1,2, \ldots \) is \( M_{x}(t)=e^{-\lambda\left(1-e^{\prime}\right)} \) aii. Show that it's mean and variance are \( \lambda \) b. Supposed the number of customers that enter a bank in an hour is a Poisson random variable and supposed that \( P(X=0)=0.05 \) i. Determine the mean and variance of \( X \)
Probability Jan 30, 2025
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