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Example class 7

# Example class 7 - are called the mean and standard...

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The University of Hong Kong Department of Statistics and Actuarial Science STAT1301 Probability and Statistics I Example class 7 Random Variables Revision Continuous Random Variables: Random variable takes on a continuum of values rather than a finite or countable infinite number. Density function Cumulative Distribution Function: The Probability that X falls in the interval (a, b) is the area under the density function between and , a) Exponential Density Function Exponential distribution is often used to model lifetime or waiting time. Suppose we consider modeling the lifetime of an electronic component, the component has lasted a length of time s, the probability that it will last t more time is The probability that the unit will last t more time does not depend on s. The exponential distribution is consequently said to be memory-less . b) Normal Distribution Central Limit Theorem : if X is a sample of a large number of independent random variables, X is approximately normally distributed, and its density function is defined, The parameters μ and σ

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Unformatted text preview: are called the mean and standard deviation of the normal density. Exercise 1. The loss due to a fire in a commercial building is modeled by a random variable X having density function, What is the probability of having a loss exceeding 5? 2. The time until failure of a certain electronic device follows the probability density function , t = number of days Determine the probability that the device will fail within 30 days. 3. . Determine the density function of Y = a X + b for some constants a, b with a>0. 4. An actuary is designing a new game for a casino. Out of the 100 games, the mean of the expected profit of the casino on the new game is 5, and the variance is 100. Using a normal approximation, a) Determine the probability that the casino’s profit exceeds 10. b) Determine the probability that the casino’s loss exceeds 15. . c) Determine the probability that the casino’s profit range from loss of 20 to profit of 15 out of the 100 games. . d) If the casino is certain of making profit of probability 5%, what is the expected profit?...
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