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The random variable Z has a Normal distribution with mean 0 and variance 1. Show that...
Exercise 2. Let consider a normally distributed random variable Z with mean 0 and variance 1. Compute (a) P(Z < 1.34). (b) P(Z > -0.01). (c) the number k such that P(Z <k) = 0.975.
1. Let X1, ..., Xn be random sample from a distribution with mean y and variance o2 < 0. Prove that E[S] So, where S denotes sample standard deviation. 10 points
Suppose Z is a standard normal random variable. (See problem.) If P(-z<z<z) 0.796, find Question 1 Find P(-2.46 <Z<-0.98) Question 2
29. Let Z be a standard normal random variable. (a) Compute the probability F(a) = P(2? < a) in terms of the distribution function of Z. (b) Differentiating in a, show that Z2 has Gamma distribution with parameters α and θ = 2.
4.3.2 The cumulative distribution func- tion of random variable X is 0 r<-1, Fx (x) = (z + 1)2-1 x < 1, r1 Find the PDF fx(a) of X
Let the random variable Z follow a standard normal distribution. Find P(-2.35 < Z< -0.65). Your Answer:
Assume Z is a random variable with a standard normal distribution and c is a positive number. If P(Z > c) = 0.25, then PC – c< < c) = 0.5. O True OFalse Exactly 50% of the area under the normal curve lies to the left of the mean. O True OFalse If X represents a random variable coming from a normal distribution and P(X < 5.2) = 0.5, then P(X > 5.2) = 0.5. O True O False
2x 0<x<1 Let X be a continuous random variable with probability density function f(x)= To else The cumulative distribution function is F(x). Find EX.
Show steps, thanks! 2.5.9. The random variable X has a cumulative distribution function 0, forx<0 F(x) for x > 0. for x > , 1+x2" · Find the probability density function of X.
he cumulative distribution function (cdf), F(z), of a discrete ran- om variable X with pmf f(x) is defined by F(x) P(X < x). Example: Suppose the random variable X has the following probability distribution: 123 45 fx 0.3 0.15 0.05 0.2 0.3 Find the cdf for this random variable