University:
McMaster University
Course:
STATS 1LL3 | Introduction to Probability and Statistics
Academic year:

2023
Views:

338

Pages:

5
Author:

Sabrina S.

Video 13: Normals Distribution Die tossing example E(x)= pixi 30 tosses 21 6 3.5 where x ∈ {1,2,3,4,5,63 X = sample mean On excel this 100 Simulate tossing a times coin 30 times, then repeat throws 1 2 3 1 2 3 100 = = RANDBETWEEN (1,6) ↓ 30 average AVERAGE (B2:B32) 100 1 min = MIN (B32: CW32) max = FREQUENCY (B32:CW32, bins) 2-6 → distribution of averages of 30 tosses across 100 repeats of the experiment = FREQUENCY (B2: CW32, bins) 2.8 3 32 Individual outcomes 3 4 S 6 3:4 3. 3.8 h 4.2 If the random variable X is the sum of a large number of random increments, then X will tend towards Normal Distribution. The Central Limit Theorem: If a random variable X is the sum of a large number of random increments, then X has a normal distribution. The normal distribution appropriate for continuous random variables It has a probability density function has 2 parameters mean (۲) middle (*) how spread out variance bell shaped and symmetric about the mean × ~ Ν (μ, στ) 2 Pr (left of p) 0.5 Pr (right of p) 0-5 ۲ the tails of distribution head asymptotically towards too - unable to integrate analytically which means cannot get exact formula for probability the way we would for others use table 1 in introstat Example in video X~N (100, 4) to calculate the probability that X lies within a given interval we convert from the units of X to a Standard Normal Distribution (2) where the mean = O and variance Z is measured in terms of away from the mean of X the number of standard deviations to convert from X to 2: Z : XN (100,4) Pr (98 < x < 102) Pr ( 98-100 2 X-100 2 Pr (-14 Z <1) = 102-100 2 1 X~N (100,4) Z~N (0,1) = 98 100 102 01 both congurent to each other Text book of the area under the curve for a probability density function is one How do we make do with tables for only Standard normal distribution? Because we have an easily proved result that the proportion of the density function that lies between the mean and a specified number of standard deviations away from the mean is always constant regardless of numerical values of the mean and standard deviation The general result the area between N(,) is the same as the for N (0,1) point X is and some point X for area between 0 and z= ズード Z tells us how many standard deviations that the away from the mean μ. Sum and Differences of independent random variables Suppose we have a number of tasks that have to be completed in sequence Suppose the time taken for each task obeys a normal distribution, each having a given mean and variance and is independent of the time token for other tasks. Obviously, the total time taken will also be a random variable What will its distribution mean and variance be? We state that total time taken will be normally distributed with mean total time equal to sum of means for each task, variances (not standard deviations). and variance equal to sum of Xi ~N(, (α) ) and if it is independent of time taken for other tasks, then the distribution of random variable where and σ' - Σ 2 IS ۶- (μ Sometimes we need to consider the difference of two independent normally distributed random variables Χι ~ Ν (μι, στ) then, letting z = X1 and X2 σ - Suppose Χε~Ν (με, σετ) we state without proof, 2 + 0,) 2 ) The mean of the random variable z if found by subtraction, but the variance is still found by addition.