# Doubt about distribution of the brownian motion

Let $$B_{t}$$ a brownian motion (stochastic process) then I know $$B_{t} -B_{s}$$ has a normal distribution with mean$$=0$$ and variance $$=t-s$$

I want to calculate the following probability: $$P(3B_{2}>4)$$

I know that if $$X \sim N(\mu,\sigma^2$$) then if $$k$$ is an constant $$kX \sim N(k \mu,k^2 \sigma^2)$$

therefore:

$$3B_{2} \sim 3N(0,2)=N(0,2*3^2)=N(0,18)$$ then $$P(3B_{2}>4)=0.41$$

but in $$P(B_{2}>4/3)$$,
$$B_{2} \sim N(0,2)$$ and $$P(B_{2}>4/3)=.25$$ I don't know what happens, should not give the same probability? on the other hand if $$3B_{2} \sim N(0,6)$$ the probability of $$P(3B_{2}>4)=0.25$$ Can andybody helpme?

Note that if $$X\sim N (\mu,\sigma^2)$$ then $$Pr(X>a)=Pr\left(\frac{X-\mu}{\sigma}>\frac{X-a}{\sigma}\right)$$. We devide by the standard deviation to obtain the desired probability using tables for the standard normal distribution.

I think what is going on is that you calculate these probabilities using variance rather than standard deviation. The standard deviation of a Brownian motion is $$\sqrt{t}$$ and of $$3B_t$$ is $$3\sqrt{t}$$. I used MatLab to compute these probabilities and I got: $$Pr(B_2>4/3)\approx 0.1729 \\ Pr(3B_2>4)\approx 0.1729$$ where to compute the first probability I assumed that $$B_2\sim N(0,2)$$ and to compute the second probability I assumed that $$3B_2 \sim N(0,18)$$. However, if in computations I pass variances instead to standard deviations then I obtain your mistake.

• Ohhh I see, thanks! :) – Jazz May 15 at 18:59
• You're welcome! – Mdoc May 15 at 19:18

We have $$\Bbb P(3B_2>4) = \Bbb P\left(\frac {B_2}{\sqrt 2}>\frac{2\sqrt 2}3\right) = \Bbb P\left(\frac {B_2}{\sqrt 2}<-\frac{2\sqrt 2}3\right) = \Phi\left(-\frac{2\sqrt 2}3\right) \ ,$$ where $$\Phi$$ is the repartition of the standard normal distribution $$N(0, 1^2)$$, since $$B_2/\sqrt 2$$ respects this distribution. This is roughly, as computed by sage:

sage: T = RealDistribution('gaussian', 1)
sage: T.cum_distribution_function( -2*sqrt(2)/3 )
0.17288929307558015