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3. 1if the random variable y has an expected value of...

# Question: 1if the random variable y has an expected value of...

###### Question details

1.If the random variable Y has an expected value of μy and a variance of  σy^2 , what are the expected value and variance of  W=aY+b ?

2. Suppose that the random variable X~N(μ,σ^2).  Then, show that  Y=(X-μ)^2/σ^2 has a chisquare (X^2)  distribution with one degree of freedom.

3.  Let X be the sample mean of a random sample of size 5 from a normal distribution with mean =0 and variance = 125. Determine c such that P(x̄<c)=0.90.

4.   If  is the sample mean of a random sample of size n from a normal distribution with mean =μ and variance = 100. Determine n so that P( μ-5< x̄  <μ+5)=0.954

5.  Suppose X1,X2,X3,...,X25 and  Y1,Y2,Y3,...,Y25  are two independent random samples from two normal distributions N(0,16) and  N(1,9), respectively.  Let and Ý denote the corresponding sample means. Compute P(x̄> Ý ).  (Consider the distribution of  x̄ -Ý .)

6.  Let and S^2=∑(i=1,n)(Xi-x̄ )^2/n be the sample mean and sample variance of a random sample of size 25 from a distribution that is N(3,100).  Evaluate P(0<x̄ <6 and 55.2<S^2<145.6).