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 ∼ N(µ, σ2 ). [Hint: consider using the mgf of the normal distribution.]
X is a random variable with a lognormal distribution and that Y = ln(X) ∼ N(µ, σ2 ). Prove that µX = e ^ (µ+ (σ^2)/2 )
5. A random variable X ∼ N (µ, σ2 ) is Gaussian distributed with mean µ and variance σ 2 . Given that for any a, b ∈ R, we have that Y = aX + b is also Gaussian, find a, b such that Y ∼ N (0, 1) Please show your work. Thanks!
6. Using the mgf, find the mean and variance of the random variable X with pdf: f(x)=
Suppose that a random variable is normally distributed with mean μ and variance σ2 and we draw a random sample of 5 observations from this distribution. What is the joint probability density function of the sample?
Let the random variable X follow a normal distribution with µ = 22 and σ2 = 7. Find the probability that X is greater than 10 and less than 17.
A random variable X is normally distributed with a mean of 121 and a variance of 121, and a random variable Y is normally distributed with a mean of 150 and a variance of 225. The random variables have a correlation coefficient equal to 0.5. Find the mean and variance of the random variable below. Av-218 (Type an integer or a decimal.) σ (Type an integer or a decimal.)
Assume that the random variable X is normally distributed, with mean µ = 50 and standard deviation σ = 7. Compute the probability P(X ≤ 58). Be sure to draw a normal curve with the area corresponding to the probability shaded.
4. Consider a continuous random variable X that is normally distributed with mean 4 and variance 10. i. Draw (as accurately as you can) the pdf of X. Carefully label axes. ii. Draw (as accurately as you can) the cdf of X. Carefully label axes. iii. At what value of x does the cdf take on the value 0.5? Label this in your diagram. iv. In the diagram of your pdf, label the area that represents the probability that X...
19. X is a normally distributed random variable with a mean of 8 and a variance of 9. The probability that x is greater than 13.62 is a. 0.9695 b. 0.0305 c. 0.87333 d. 0.1267
Let \(X\) be a normal random variable with mean \(\mu\), variance \(\sigma^{2}\), pdf$$ f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} $$and mgf \(M(t)=e^{\mu t+\frac{1}{2} \sigma^{2} t^{2}}\)(a) Prove, by identifying the moment generating function of \(a+b X\), that \(a+b X \sim\) \(N\left(a+b \mu, b^{2} \sigma^{2}\right)\)(b) Prove, by identifying the pdf of \(a+b X\) (via the cdf), that \(a+b X \sim N(a+\) \(\left.b \mu, b^{2} \sigma^{2}\right)\)