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Thread: Probability and Averages

  1. #1
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    Probability and Averages

    The prompt is this.

    "The prices of power tools manufactured by Whammo are known to be normally distributed with a mean of $1200 and a standard deviation of $220. Suppose that I am the buying agent for my company and I need to buy 10 power tools from Whammo. If each tool is selected randomly what is the probability that I will be able to keep the average price for the power tools below $1250? "

    I have determined that probability of a single purchase by doing (1250-1200)/220 = .23 and from there I used a Z table to find the single probability is 0.591

    My problem is I do not know how (and have not been able to find) the correct formula to find the average when it is applied 10 times. Appreciate any help anyone can provide.

  2. #2
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    Re: Probability and Averages

    In short: If X_i \stackrel{\text{i.i.d.}} {\sim} \mathcal{N}(\mu, \sigma^2), then

    \bar{X} = \frac {1} {n} \sum_{i=1}^n X_i \sim \mathcal{N}\left(\mu, \frac {\sigma^2} {n}\right)

    In a more general framework:
    - Every affine transformation of a multivariate normal random vector also follows a multivariate normal distribution
    - A set of independent univariate normal is jointly following a multivariate normal distribution, and their covariance matrix is just a diagonal matrix
    - Univariate normal is a special case of the multivariate normal distribution

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