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Random-Variable Questions
Random-Variable Questions
Let ๐ be a standard normal random variable.
Calculate Var(๐) if ๐ represents the outcome when a fair die is rolled.
Assume that a continuous random variable ๐ that is normally distributed with a mean of 100 and a standard deviation of 16. Find the probability that ๐ being a. greater than 50, b. less than 30
A random variable X is lognormal if ln X, the natural logarithm of X, is normally distributed. Find the mean and variance of a lognormal random variable with ln X โผ N(ยต, ฯ2 ). [Hint: consider using the mgf of the normal distribution.]
The Bernoulli random variable takes values 0 and 1, and has probability function f(x) = (p^x) [(1 โ p)^(1โx)]
Let X be a discrete random variable with probability function f(x). Prove that E[a + b g(X) + c h(X)] = a + bE[g(X)] + cE[h(X))], where g and h are functions, and a, b and c are constants.
Let a random variable X have the p.f. f(x) = c(|x| + 1)2 for x = 0, ยฑ1. Find the constant c so that f(x) is a valid probability function. Then, find E(X), E(X2 ) and var(X).
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